Triangles Questions

We provide triangles practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on triangles skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of triangles Questions

Question No Questions Class
1 In a right angled ( triangle mathrm{ABC}, mathrm{AC} perp ) the
hypotenuse BC then ( A D^{2} ) is
( mathbf{A} cdot A C times C D )
B. ( A C times B A )
c. ( D C times B D )
D. None of these
10
2 In a squared sheet, draw two triangles of equal area such that the triangles are congruent.comment about perimeter. 9
3 Show that the diagonals of a rhombus divide it into four congruent triangles. 9
4 66. In a quadrilateral ABCD, with un-
equal sides if the diagonals AC
and BD intersect at right angles,
then
(1) AB2 + BC2 = CD2 + DA2
(2) ABP + CD = BC2 + DA2
(3) AB2 + AD = BC + CD2
(4) AB2 + BC = 2(CD? + DA?)
10
5 In the adjoining figure ( triangle P Q R sim )
( triangle boldsymbol{T} boldsymbol{S} boldsymbol{R} )
ldentify the corresponding vertices, corresponding sides and their ratios.
10
6 ABCD is a trapezium. Further if ( mathrm{CD}=4.5 ) ( mathrm{cm} ; ) find the length of ( 2 mathrm{AB} ) in ( mathrm{cm} ) 10
7 n given figure, ( angle C A B=90^{circ} ) and
( A D perp B C . ) If ( A C=75 mathrm{cm} . A B=1 mathrm{m} ) and
( B D=1.25 mathrm{m}, ) find ( A D )
10
8 A ( 15 mathrm{m} ) long ladder reached a window 12
( mathrm{m} ) high from the ground. On placing it against a wall at a distance ( x ) m. Find ( x )
10
9 State true or false:
In ( triangle A B C, A B=B C=C A=2 a ) and
( A D ) is perpendicular to side ( B C, ) then ( A D=2 a sqrt{3} )
A. True
B. False
10
10 A vertical stick ( 12 mathrm{m} ) long casts a
shadow ( 8 mathrm{m} ) long on the ground. At the
same time a tower casts the shadow of
length ( 40 mathrm{m} ) on the ground. Determine the height of the tower
10
11 f ( Delta A B C ) and ( Delta X Y Z ) are congruent
then ( Delta A B C ldots ldots . Delta X Y Z )
( A cdot cong )
B. ( = )
( c cdot approx )
D.
9
12 State which pairs of triangles in Fig. are
similar. Write the similarity criterion
used by you for answering the question and also write the pairs of similar
triangles in the symbolic form:
10
13 Determine by choosing the best option for the similarly of triangles theorem. ( D E | B C . ) Both ( triangle A D E & triangle A B C ) are
similar by:
A. SAS
B. SSS
( c cdot A A A )
D. AAS
10
14 From a point ( Q, ) the length of the
tangent to a circle is ( 24 mathrm{cm} ) and the
distance of ( Q ) from the centre is ( 25 mathrm{cm} ) The radius of the circle is
10
15 ( A B=6.3 mathrm{cm}, mathrm{EC}=11.0 mathrm{cm}, mathrm{AD}=0.8 mathrm{cm} ) and
( A E=1.6 mathrm{cm} )
Enter 1 if true or 0 if false.
10
16 n fig. ( mathbf{7 . 8}, boldsymbol{O A}=boldsymbol{O B} ) and ( boldsymbol{O D}=boldsymbol{O C} )
Show that (i) ( Delta A O D cong Delta B O C ) and (ii)
( boldsymbol{A} boldsymbol{D} | boldsymbol{B} boldsymbol{C} )
9
17 State and prove Basic proportionality (Thales) theorem. 10
18 In the figures, sides ( X Y ) and ( Y Z ) and
median XA of a triangle XYZ are
propotional to sidesDE,EF and median
DB of ( triangle D E F ). Show that ( Delta X Y Z sim )
( Delta D E F )
10
19 In a triangle ( A B C, ) medians ( A D ) and ( B E )
are drawn. If ( mathrm{AD}=4, angle D A B=frac{pi}{6} ) and
( angle A B E=frac{pi}{3}, ) then the area of ( Delta A B C ) is
A ( cdot frac{8}{3} )
в. ( frac{16}{3} )
c. ( frac{32}{3 sqrt{3}} )
D. ( frac{64}{3} )
10
20 Two triangles are ……. if two sides and included angle of one triangle are equal to two sides and included angle of the other triangle.
A. congruent
B. unequal
c. equilateral
D. none of these
9
21 State whether the shapes in the given pair are similar. If they are similar enter
else if not similar then enter 0
10
22 Which of the following statements is true, if ( Delta P Q R cong Delta L M N ? )
( mathbf{A} cdot P Q=M N )
в. ( Q R=L N )
c. ( P R=L M )
D. ( Q P=M L )
9
23 n Fig., ( A B C ) and ( A M P ) are two right
triangles, right angled at ( B ) and ( M )
respectively. Prove that:
( triangle A B C sim triangle A M P )
10
24 Which one of the four trapezoids is not
similar to the other three?
( A cdot A )
B. B
( c cdot c )
( D )
10
25 n given figure, express ( x ) in terms of ( a, b )
and ( c )
10
26 The areas of two similar triangles are ( 100 mathrm{cm}^{2} ) and ( 64 mathrm{cm}^{2} ). If the median of
greater side of first triangle is ( 13 mathrm{cm} ) find the corresponding median of the other triangle.
A . ( 20 mathrm{cm} )
B. ( 15 mathrm{cm} )
c. ( 10 mathrm{cm} )
D. ( 16 mathrm{cm} )
10
27 61. If G is the centroid of A ABC and
A ABC = 48 cm², then the area
of A BGC is
(1) 32 cm (2) 8 cm
(3) 16 cm2 (4) 24 cm
9
28 If ( Delta mathbf{A B C} sim )
( Delta P Q R ) then find value of ( y+3 )
10
29 In the given figure AE is bisector of
( angle B A C ) and of ( angle B D C . ) show that
( Delta A B D cong A C D ) and hence ( B D=C D )
10
30 n given figures the two triangles are
( operatorname{similar} Delta A B C sim Delta D E F ) if ( E D=7 ) find
the length of ( boldsymbol{A C} )
( A )
B. 14
( c )
( D cdot 2 )
10
31 In two similar triangles ( A B C ) and ( P Q R ), if their corresponding altitudes ( A D ) and ( P s ) are in the ratio ( 4: 9, ) find the ratio of the
areas of ( triangle A B C ) and ( triangle P Q R )
A . 16: 81
B. 9: 16
c. 81: 16
D. 16: 9
10
32 Which among the following is/are not
correct?
This question has multiple correct options
A. The ratios of the areas of two similar triangles is equal to the ratio of their corresponding sides.
B. The areas of two similar triangles are in the ratio of the corresponding altitudes.
C. The ratio of area of two similar triangles are in the ratio of the corresponding medians.
D. If the areas of two similar triangles are equal, then the triangles are congruent
10
33 Triangle ( A B C ) is similar to triangle
PQR. Find ( P Q )
10
34 ratio ( boldsymbol{O P}: boldsymbol{O} boldsymbol{D}=boldsymbol{m}: boldsymbol{n} . ) Find ( boldsymbol{m}+boldsymbol{n} ) 10
35 If diagonal of a rectangle is ( 26 mathrm{cm} ) and one side is ( 24 mathrm{cm} ).Find the other side? 10
36 If ( Delta A B C cong Delta F E D ) under the
correspondence ( boldsymbol{A B C} leftrightarrow boldsymbol{F E D}, ) write
all the corresponding congruent parts of the triangles.
9
37 In the figure given below, ( angle boldsymbol{P}=angle boldsymbol{R} boldsymbol{T} boldsymbol{S} )
Prove that ( triangle R P Q sin triangle R T S )
10
38 In ( triangle A B C, angle C=90^{circ} . ) If ( B C=a, A C= )
( b ) and ( A B=c, ) find ( c ) when ( a=8 c m )
and ( b=6 mathrm{cm} )
( mathbf{A} cdot 10 mathrm{cm} )
в. ( 14 mathrm{cm} )
c. ( 15 mathrm{cm} )
D. ( 20 mathrm{cm} )
10
39 adod
( operatorname{In} Delta A B C, angle A=90^{circ} ) and ( A D perp B C )
Then, ( Delta A B C sim Delta D A C )
A. True
B. False
10
40 Two equilateral triangles with side ( 4 mathrm{cm} )
and ( 6 mathrm{cm} ) are ( _{–}-_{-} ) triangles.
A. similar
B. congruent
c. both
D. none of these
10
41 Which line is parallel to ( B C ? )
A ( . P Q )
3.57
( c, O R )
D ( S ) h
10
42 n the figure given below, ( Delta B C A cong )
( Delta B C D . ) Corresponding angle to ( angle D ) is
( A cdot angle B )
B. ( angle C )
( c . angle D )
0.4
9
43 State true or false:
In quadrilateral ( A B C D, ) its diagonals ( A C ) and ( B D ) intersect at point ( O, ) such
that ( frac{boldsymbol{O C}}{boldsymbol{O A}}=frac{boldsymbol{O D}}{boldsymbol{O B}}=frac{1}{3}, ) then
( triangle boldsymbol{O} boldsymbol{A} boldsymbol{B} sim triangle boldsymbol{O C D} )
A. True
B. False
10
44 62.
If the circumcentre of a triangle
lies on one of the sides then the
orthocentre of the triangle lies on
(1) one of the vertices
(2) on the same side of the trian-
gle
(3) outside the triangle
(4) strictly inside the triangle
9
45 The two triangles below are similar
( Delta A B C sim Delta E F D )
What is the value of ( angle B ? )
A . ( 80^{circ} )
B. 30
( c cdot 70^{circ} )
D. 60
9
46 I a ( triangle A B C, angle A=90^{circ}, C A=A B ) and ( D ) is a
point on AB produced prove that ( boldsymbol{D} boldsymbol{C}^{2}=boldsymbol{B} boldsymbol{D}^{2}=boldsymbol{2} boldsymbol{A} boldsymbol{B} times boldsymbol{A} boldsymbol{D} )
10
47 Find the zeros of the quadratic
polynomial ( x^{2}+7 x+12, ) and verify the
relation between the zeros and its
coefficients.
10
48 Let ( A B C ) be an equilateral triangle and
suppose KLMN be a rectangle with ( boldsymbol{K}, boldsymbol{L} ) on ( mathrm{BC}, mathrm{M} ) on ( mathrm{AC} ) and ( mathrm{N} ) on AB. If ( frac{A N}{N B}=2 ) and area of triangle BKN is ( 6, ) then area of triangle ( A B C ) is equal to
A . 48
B. 54
c. 96
D. 108
9
49 In the following figure, ( X Y ) is parallel to ( B C, A X=9 c m, X B=4.5 mathrm{cm} ) and
( B C=18 mathrm{cm} . ) Find the value of ( x y )
( A cdot 14 )
3. 12
( c .16 )
D. 18
10
50 Side ( A B ) and ( B C ) and median ( A D ) of a
triangle ( A B C ) are respectively
proportional to sides ( P Q ) and ( Q R ) and
median ( P M ) of ( P Q R ) Show that
( A B C sim P Q R )
10
51 The ratio of the corresponding sides of
the two similar triangles is 2: 3 and the
area of the smaller triangle is 64 sq.cm.
Find the area of larger triangle.
10
52 Find ( A B )
( mathbf{A} )
3
( c )
( D )
10
53 Find the value of ( frac{K N}{L M} ) from the given figure.
A
в. ( frac{2}{5} )
c. ( frac{2}{3} )
D. 5 ( overline{3} )
10
54 (i) If area ( (triangle A B C)=16 mathrm{cm}^{2}, ) area ( (triangle )
( mathrm{DEF}=25 mathrm{cm}^{2} ) and ( mathrm{BC}=2.3 mathrm{cm}, ) find EF.
(ii) If area ( (triangle A B C)=9 c m^{2}, operatorname{area}(triangle D E F) )
( =64 mathrm{cm}^{2} ) and ( mathrm{DE}=5.1 mathrm{cm}, ) find ( mathrm{AB} )
(iii) If ( A C=19 mathrm{cm} ) and ( mathrm{DF}=8 mathrm{cm}, ) ratio of
the area of two triangles.
(iv) If area ( (triangle A B C)=36 mathrm{cm}^{2}, ) area ( (triangle )
DEF) ( =64 mathrm{cm}^{2} ) and ( mathrm{DE}=6.2 mathrm{cm}, ) find ( mathrm{AB} )
(v) If ( A B=1.2 mathrm{cm} ) and ( mathrm{DE}=1.4 ) find the ratio
of the areas of triangle ( A B C ) and DEF.
10
55 In the given figure, the line segment
( X Y ) is parallel in ( A C ) of ( Delta A B C ) and is
divided the triangle into two parts of equal areas then ( frac{A X}{A B}=frac{sqrt{2}-1}{sqrt{2}} )
A . True
B. False
10
56 58. The angles of two right angled
triangles (AABC and APQR) are
45° and 60° respectively and
ZB and 29 are right angles
and AB = 5 cm, then PQ will
be equal to
(1) 7.5 cm (2) 5.0 cm
(3) 6.7 cm
(4) None of these
9
57 In the figure, ( C D ) and ( R S ) are
respectively the medians of ( triangle A B C ) and
( triangle P Q R ) If ( triangle A B C sim triangle P Q R, ) prove that
(i) ( triangle A D C sim triangle P S R )
(ii) ( frac{C D}{R S}=frac{A B}{P Q} )
10
58 In a triangle ( P Q R, L ) and ( M ) are two points
on the base ( Q R, ) such that ( angle L P Q= )
( angle Q R P ) and ( angle R P M=angle R Q P . ) Prove
that
( boldsymbol{P Q}^{2}=boldsymbol{Q} boldsymbol{R} times boldsymbol{Q} boldsymbol{L} )
10
59 ( A B=x c m, B C=(4 x+4) mathrm{cm} ) and ( A C=(4 x) )
+5) ( mathrm{cm} . ) Find ( mathrm{AC} )
10
60 In the figure given below, if DE ( | mathrm{BC} )
then the value of ( x ) equals to:
( A cdot 3 mathrm{cm} )
B. ( 2 mathrm{cm} )
( c cdot 4 mathrm{cm} )
D. ( 6.7 mathrm{cm} )
10
61 Two plane figures are said to be
congruent if they have
A. The same size
B. The same shape
c. The same size and the same shape
D. none
9
62 Triangles ( A B C ) and DEF are similar. If their areas are ( 64 mathrm{cm}^{2} ) and ( 49 mathrm{cm}^{2} ) and
if ( A B ) is ( 7 mathrm{cm}, ) then find the value of DE.
( A cdot 8 mathrm{cm} )
в. ( frac{49}{8} mathrm{cm} )
c. ( frac{8}{49} mathrm{cm} )
D. ( frac{64}{7} mathrm{cm} )
10
63 In the figure, ( triangle A B C ) is right angled at
( B . D ) is any point on ( A B )
( operatorname{seg} D E perp operatorname{sid} e A C . ) If ( A D= )
( 6 mathrm{cm}, A B=12 mathrm{cm} A C=18 mathrm{cm} . ) Find
( A E )
( A cdot 5 c m )
в. 3 ст
( c .4 c m )
( 0.6 mathrm{cm} )
10
64 Which of the following statements is incorrect when ( Delta P Q R cong Delta L M N ? )
( mathbf{A} cdot angle P=angle L )
в. ( angle Q=angle M )
c. ( angle R=angle N )
D. none of these
9
65 Give any two real-life examples of
congruent shapes.
9
66 Draw a ( triangle A B C ) in which ( B C= )
( mathbf{6} c boldsymbol{m}, boldsymbol{A} boldsymbol{B}=mathbf{4} boldsymbol{c m} ) and ( boldsymbol{A} boldsymbol{C}=mathbf{5} boldsymbol{c m} )
Draw a triangle similar to ( triangle A B C ) with its sides equal to ( left(frac{3}{4}right)^{t h} ) of the corresponding sides of ( triangle boldsymbol{A B C} )
10
67 Fill in the blanks.
The perimeters of two similar triangles are ( 25 mathrm{cm} ) and ( 15 mathrm{cm} ) respectively. If one side of the first triangle is ( 9 mathrm{cm}, ) then corresponding side of second triangle is
10
68 The corresponding altitudes of two
similar triangles are ( 6 mathrm{cm} ) and ( 9 mathrm{cm} )
respectively. Find the ratio of their
areas.
10
69 Instead of walking along two adjacent sides of a rectangular field,a boy book a short – cut along the diagonal of the field and saved a distance equal to ( 1 / 2 )
the longer side. The ratio of the shorter side of the rectangle to the longer side
was:
( A )
B.
( c cdot frac{1}{4} )
0.3
( E )
10
70 ( triangle A B C sim triangle D E F ) and their perimeters
are ( 32 mathrm{cm} ) and ( 24 mathrm{cm} ) respectively. If
( A B=10 mathrm{cm}, ) Find ( D E )
10
71 fin a ( triangle D E F, G H | E F ) and ( D G )
( boldsymbol{E G}=boldsymbol{2}: boldsymbol{3} ) then the value of
( frac{operatorname{ar}(triangle D G H)}{operatorname{ar}(triangle D E F)} )
A ( cdot frac{2}{15} )
B. ( frac{4}{15} )
c. ( frac{4}{25} )
D. ( frac{2}{25} )
10
72 61. In a triangle ABC, ZA = 90°, 2C
= 55°, AD IBC. What is the
value of ZBAD?
(1) 35° (2) 60°
(3) 45°
(4) 55°
9
73 Two triangles ( A B C ) and ( P Q R ) are ( operatorname{similar}, ) if ( B C: C A: A B=1: 2: 3 )
then ( frac{Q R}{P R} )
A ( cdot frac{3}{2} )
B. ( frac{1}{2} )
c. ( frac{1}{sqrt{2}} )
D. ( frac{2}{3} )
10
74 n ( triangle boldsymbol{R} boldsymbol{S T} ), line ( boldsymbol{P Q} | ) segST ( , boldsymbol{R}-boldsymbol{P}-boldsymbol{S} )
and ( boldsymbol{R}-boldsymbol{Q}-boldsymbol{T} . ) If ( boldsymbol{R} boldsymbol{P}=boldsymbol{4}, boldsymbol{P} boldsymbol{S}= )
( 8, R Q=3, ) then find ( Q T )
10
75 Is similarity of triangles different from similarity of polygons? 10
76 ( O^{prime} ) is a point inside of ( triangle A B C . ) The
bisector of ( angle A O B, angle B O C, angle C O A ) meet
the sides ( A B, B C ) and ( C A ) in points
( D, E ) and ( F ) respectively, then
( boldsymbol{A} boldsymbol{D} cdot boldsymbol{B} boldsymbol{E} cdot boldsymbol{C} boldsymbol{F}= )
A. ( D B . E C . F A )
в. ( A D . E C . F A )
c. ( D B . B ) E.FA
D. DB.BE.CF
10
77 In the given figure, ( triangle A B C ) is an isosceles right angled triangle with
( angle B=90 ) such that ( P Q perp A C, S T perp A C )
where Plies on AB and S lies on BC. Then
prove that ( triangle A Q P sim triangle C T S )
10
78 59. The ratio of the areas of the
incircle and the circumcircle
of a square is :
(1) 1:2 (2) 2:3
(3) 3:4 (4) 4:5
10
79 n the given figure, ( P S=P R . angle T P S= )
( angle Q P R ) Then ( P T=P Q )
4 True
3. Fals
10
80 The corresponding sides of two similar triangles are ( 4 mathrm{cm} ) and ( 5 mathrm{cm} ) respectively, find the ratio of the area of the first triangle to the area of the 2 nd triangle: 10
81 There were three circular tracks made
in a park having the same middle point but their radii was different. These
tracks will be called
A. Not similar
B. Similar
c. congruent
D. All of the above
10
82 Check whether the following pairs of
triangles are congruent. If they are
congent, state the congruence criterion.
9
83 In the given figure, ( triangle A B C ) is right
angled at ( B ). If ( B D perp A C ), which of the
following is/are true?
A. ( triangle A B C sim triangle A D B )
B. ( triangle A B C sim triangle B D C )
c. ( triangle A D B sim triangle B D C )
D. ( triangle A B C sim triangle D B C )
10
84 Let ( triangle A B C sim triangle D E F ) and their areas
be ( 64 mathrm{cm}^{2} ) and ( 121 mathrm{cm}^{2} ) respectively. If
( boldsymbol{E F}=mathbf{1 5 . 4} boldsymbol{c m}, ) find ( boldsymbol{B C} )
10
85 Triangles ( A B C ) and ( P Q R ) are similar
Find the angle ( B )
( A cdot 30^{circ} )
B. ( 60^{circ} )
( c cdot 90^{circ} )
0.70
10
86 If the vertices of an equilateral triangle
have integral co-ordinates, then
A. Such a triangle is not possible
B. The area of the triangle is irrational
C. The area of the triangle is an integer
D. The area of the triangle is rational but not an integer
9
87 Two quadrilaterals, a square and a rectangle are not similar as they ( ldots ). in shape as well as size.
A . Differ
B. Are same
c. Do not siffer
D. Angles also differ
10
88 ( triangle A B C ) is similar to ( triangle X Y Z ) by ( S A S ) similarity. If in ( triangle A B C A B= )
( mathbf{1 2}, boldsymbol{B C}=mathbf{8}, angle boldsymbol{B}=mathbf{6 0} )
and in ( triangle boldsymbol{X} boldsymbol{Y} boldsymbol{Z} boldsymbol{X} boldsymbol{Y}=mathbf{3}, angle boldsymbol{Y}=mathbf{6 0} . ) Find
the value of ( Y Z )
A . 15
B . 20
( c cdot 4 )
D. 2
10
89 60. The number of points in the plane
of a triangle which are equidis-
tant from the sides of the trian-
gle is
(1) 1
(2) 2
(3) 3
(4) 4
9
90 In the figure, find the area of ( triangle boldsymbol{P Q R} )
and the height ( P S ).
10
91 In figure ( angle mathbf{1}=angle mathbf{2} ) and ( triangle boldsymbol{N} boldsymbol{S} boldsymbol{Q} cong )
( triangle M T R, ) then prove that ( triangle P T S sim )
( triangle boldsymbol{P R Q} )
10
92 Without drawing exact triangles, state, giving reasons, whether the given pairs of triangle are congruent or not:
In ( Delta A B C ) and ( Delta P B C ; A B= )
( boldsymbol{B P}, boldsymbol{A C}=boldsymbol{P C} )
10
93 n ( triangle A B C, angle A B C=90^{circ}, B D perp A C ) If
( boldsymbol{B} boldsymbol{D}=boldsymbol{8} mathbf{c m}, boldsymbol{A} boldsymbol{D}=boldsymbol{4} mathrm{cm}, ) find ( boldsymbol{C} boldsymbol{D} )
10
94 Find the length of ( B D ) in the given
figure
( mathbf{A} )
B.
( c cdot 12 )
0.15
10
95 The altitude of an equilateral triangle of side lenght of ( 2 sqrt{3} mathrm{cm} ) is:
A ( cdot frac{sqrt{3}}{2} c )
в. ( frac{1}{2} c m )
c. ( frac{sqrt{3}}{4} c )
D. 3 ст
10
96 ( A B C ) is a triangle whose altitudes ( B E )
and ( C F ) to sides ( A C ) and ( A B )
respectively, are equal. Which of these conditions is not required to prove ( triangle A B E cong triangle A C F ? )
A ( . angle B=angle C )
B. ( angle B A E=angle F A C )
c. ( angle A F C=angle A E B )
D. ( B E=C F )
9
97 In triangle ( A B C ), the altitude from ( A ) to
( B C ) meets ( B C ) at ( D ) and the altitude
from ( B ) to ( C A ) meets ( A D ) at ( H . ) If ( A D= )
( 4 mathrm{cm}, B D=3 mathrm{cm} ) and ( C D=2 mathrm{cm} ) and
if ( frac{A B}{B D}=frac{A H}{H D}, ) then the length of ( H D )
is
( ^{text {A } cdot frac{sqrt{5}}{2} mathrm{cm}} )
в. ( frac{3}{2} mathrm{cm} )
c. ( sqrt{5} mathrm{cm} )
D. cn
10
98 PQR is a triangle. S is a point on the side
QR of ( Delta ) PQR such that ( angle mathrm{PSR}=angle mathrm{QPR} / )
Given ( boldsymbol{Q} boldsymbol{P}=mathbf{8 c m}, P mathrm{R}=mathbf{6 c m} ) and ( mathbf{S R}=mathbf{3} )
( mathrm{cm} )
(I) Prove ( Delta mathrm{PQR} sim Delta ) SPR
(II) Find the length of QR and PS
areaof ( Delta P Q R )
(III) ( frac{d r e a o f Delta S P R}{a r e a o f} )
10
99 87. The ratio of the areas of two isos-
celes triangles having the same
vertical angle (i.e. angle between
equal sides) is 1:4. The ratio of
their heights is
(1) 1:4 (2) 2:5
(3) 1:2 (4) 3:4
10
100 n figure ( D ) and ( E ) are the mid points of
sides ( A B ) and ( A C ) respectively of ( triangle A B C )
Find ( angle E D B )
( A cdot 110 )
( 3 cdot 120 )
( c cdot 70^{circ} )
( D cdot 80 )
10
101 Prove that if the areas of two similar
triangle are equal, then the triangles
are congruent
9
102 Fill in the blank: ( angle C cong )
( A cdot Z )
3. ( angle Q )
( c . angle R )
D. ( angle A )
10
103 66. AABC and ADBC are two isosce-
les triangles formed on opposite
side at the same base BC and Z
A= 80º and ZD = 70° then ZABD
will be equal to
(1) 115° (2) 150°
(3) 75° (4) 105°
9
104 The areas of two similar triangles are
( 121 mathrm{cm}^{2} ) and ( 81 mathrm{cm}^{2} ) respectively. Find the ratio of their corresponding heights.
A ( cdot frac{11}{9} )
в. ( frac{10}{9} )
( c cdot frac{9}{11} )
D. ( frac{9}{10} )
10
105 n the figure, ( angle P Q S=angle P R T ) and
( mathrm{QS}=mathrm{TR} . ) then ( Delta P Q S cong Delta P R T mathrm{by} )
congruence postulate
( A cdot operatorname{sas} )
B. sss
( c . ) RHS
D. AAS
9
106 n the figure of ( Delta A B C, D E | A B )
( f A D=2 x, D C=x+3, B E=2 x-1 ) and
( C E=x, ) then find the value of ( x )
10
107 In ( triangle A B C, angle A B C=90^{circ} ) and BM is the
altitude. If ( A M=16 ) MC Prove that ( A B= )
( 4 mathrm{BC} )
10
108 In the adjoining figure ( D E | B C ), if
( A C=18 c m, A D=6 c m, A B=12 c m )
find ( boldsymbol{E C} ) ?
10
109 ff ( Delta D E F equiv Delta B C A, ) write the part(s) of
( Delta B C A ) that correspond to ( angle F )
9
110 ( f(-A) ) and ( C B ) are congruent, ( A C=12 )
then find ( B C ).
4. 12
3.10
( c )
2.2
10
111 From the following data, state if ( triangle A B C sim triangle D E F ) or not
( angle B=65^{circ}, angle C=82^{circ}, angle D=33^{circ}, angle F= )
( mathbf{6 5}^{circ} )
10
112 Assertion: Two ( Delta ) s are said to be
congruent if two sides and an angle of the one triangle are respectively equal to the two sides and an angle of the other triangle.

Reason: Two ( Delta ) s are congruent if two sides and the included angle of the one
triangle are equal to the corresponding two sides and included angle of the other triangle.

Two statements A and R are given above. Which of the following statements is correct?
A. A is false and R is the correct explanation of A
B. A is true and R is the correct explanation of a A
c. ( A ) is false and ( R ) is true
D. None of these

9
113 n Fig. ( D ) is a point on hypotenuse ( A C )
of ( triangle A B C, ) such that
( B D perp A C, D M perp B C ) and ( D N perp A B )
Prove that:
(i) ( D M^{2}=D N cdot M C )
(ii) ( D N^{2}=D M cdot A N )
10
114 n given figure ( A B C D ) is a trapezium in
which ( A B | C D ) and ( A D=B C . ) Show
that
(i) ( angle boldsymbol{A}=angle boldsymbol{B} )
(ii) ( angle C=angle D )
(iii) ( triangle A B C cong triangle B A D )
(iv) diagonal ( A C=operatorname{diagonal~} B D )
9
115 If ( D E | B C ) in ( triangle A B C, A D= )
( 1.5 mathrm{cm}, B D=3 mathrm{cm} ) and ( A E=1 mathrm{cm}, )
then find ( E C(text { in } mathrm{cm}) . )
10
116 n the above figure find the ratio
between the ( triangle A O B ) and ( triangle C O D, ) if ( A B= )
( mathbf{3} C D )
10
117 Prove that “In a trapezium, the line
joining the mid points of non-parallel
sides is
(i) parallel to the parallel sides and
(ii) Half of the sum of the parallel sides”
10
118 Find the similarity statement
A. ( Delta T S R sim Delta P Q R )
B. ( Delta T P S sim Delta Q R T )
c. ( Delta R P Q sim Delta T R P )
D. ( Delta S R Q sim Delta P Q S )
10
119 71. Two medians AD and BE of AABC
intersect at G at right angles. If
AD = 9 cm and BE = 6 cm, then
the length of BD, in cm, is
(1) 10
(2) 6
(3) 5
(4) 3
9
120 ( Delta A P B ) is similar to ( Delta C P D )
State whether the above statement is
true or false.
10
121 n which of the following cases the pairs of triangles are similar? Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in
the symbolic form
10
122 State True or False.
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar.
A. True
B. False
10
123 In the following figure, ( M ) is midpoint of
BC of a parallelogram ABCD. DM intersects the diagonal ( A C ) at ( P ) and
AB produced at E. Then
( mathbf{A} cdot P E=3 P D )
B. ( P E=2 P D )
c. ( P E=P D )
( D . P E=4 P D )
10
124 70. ABC is an equilateral triangle and
O is its circumcentre, then the
ZAOC is
(1) 100° (2) 110°
(3) 120° (4) 130°
9
125 If the areas of two similar triangles are equal then the triangles:
A. are congruent
B. have equal length of corresponding sides
c. (A) and (B)
D. None of these
10
126 65. Two chords AB and CD of a cir-
cle intersect at P inside the cir-
cle and PA = 8 cm, AB = 14 cm,
PC = 4, cm then find the length
of CD
(1) 6 cm (2) 10 cm
(3) 12 cm (4) 16 cm
10
127 68. If a, b and care the sides of a
triangle and a + b + c = ab +
bc + ca, then the triangle is
(1) right-angled
(2) obtuse-angled
(3) equilateral
(4) isosceles
10
128 ( ln Delta A B C, ) a line is drawn parallel to
( B C ) to meet sides ( A B ) and ( A C ) in ( D ) and
( E ) respectively. If the area of the ( Delta A D E ) is ( frac{1}{9} ) times area of the ( Delta A B C, ) then the value of ( frac{A D}{A B} ) is equal to:
A ( cdot frac{1}{3} )
B.
( c cdot frac{1}{5} )
D.
10
129 n the given figure, the value of ( x text { (in } c m) )
is :
4
B.
( c )
( D )
10
130 Find the unknown length ( x ) in the
following figures:-
10
131 ( angle A D C=angle B C D ) 9
132 If the ratio of the perimeter of two
similar triangles is ( 4: 25, ) then find the
ratio of the areas of the similar
triangles
10
133 Two sides of a triangle are ( 64 mathrm{m} ) and 48
m. If height of the triangle corresponding to ( 48 mathrm{m} ) side is ( 6 mathrm{m} ), then the height of the triangle corresponding to ( 64 mathrm{m} ) side is ( 4.5 mathrm{m} )
If true then enter 1 and if false then
enter 0
10
134 If all the three altitudes of a triangle are equal, the triangle is equilateral. State whether the above statement is
true or false.
A. True
B. False
10
135 The perimeter of two similar triangle
are ( 30 mathrm{cm} ) and ( 20 mathrm{cm} . ) If one side of first
triangle is ( 12 mathrm{cm} ) determine the corresponding side of second triangle.
A. ( 8 mathrm{cm} )
в. ( 4 mathrm{cm} )
( mathrm{c} .3 mathrm{cm} )
D. ( 16 mathrm{cm} )
10
136 If ( A B C ) and ( B D E ) are two equilateral
triangles such that ( D ) is the mid-point then find ( a r(triangle A B C): a r(triangle B D E) )
10
137 61. Let a, B. y be the three angles of
a triangle ABC such that a +B<
7. Then A ABC is
(1) Right angled
(2) Acute angled
(3) Obtuse angled
(4) Isosceles
9
138 67. The an
The area of the incircle of an equi-
lateral triangle of side 42 cm is
(Take n = 27
(1) 231 cm2 (2) 462cm2
(3) 22/3 cm2 (4) 924 cm2
10
139 Identify the postulate, based on which the given pair of triangle can be said
similar?
A. SAS similarity postulate
B. AAA similarity postulate
c. sss similarity postulate
D. AAS similarity postulate
10
140 In a squared sheet, draw two triangles of equal area such that the triangles are
not congruent. What can you say about their perimeters?
9
141 Give any two real-life examples for
congruent shapes.
9
142 If corresponding angles of two triangles are equal, then they are known as
A . Equiangular triangles
B. Adjacent angles
c. supplementary angles
D. complementary angles
10
143 Which of the following is true?
A. The ratio of sides of two similar triangles is same as the ratio of their corresponding altitudes.
B. The ratio of sides of two similar triangles is same as the ratio of their corresponding perimeters.
C. The ratio of sides of two similar triangles is same as the ratio of their corresponding area
D. The ratio of sides of two similar triangles is same as the ratio of their corresponding medians.
10
144 n fig, If ( Delta A B E cong Delta A C D, ) show that
( Delta A D E ) is similar to ( Delta A B C )
10
145 56. In a AABC, AB? + AC = BC?
and BC = V2 AB, then ZABC
is :
(1) 30° (2) 45°
(3) 60°
(4) 90°
10
146 If ( Delta A B C equiv Delta F E D ) under the
correspondence ( A B C leftrightarrow F E D, ) write all the corresponding congruent parts of the triangles.
9
147 Triangle ( A ) has a base of ( x ) and a height of ( 2 x . ) Triangle ( B ) is similar to triangle ( A ) and has a base of ( 2 x ). What is the ratio
of the area of triangle ( A ) to triangle ( B ? )
A . 1: 2
B. 2:
( c cdot 2: 3 )
D. 1: 4
10
148 Which one of the four hexagons is not
similar to the other three?
( A )
B. B
( c cdot c )
D.
10
149 Given line ( A B ) is parallel to line ( C D )
( angle A G E ) and ( angle B G H ) are:
A. Non-congruent
B. congruent
c. Supplementary
D. None of these
9
150 State whether the following triangles
are congruent or not? Give reasons for
your answer
9
151 In ( triangle boldsymbol{L} M boldsymbol{N}, triangle boldsymbol{L}=boldsymbol{6} boldsymbol{0}^{circ}, angle boldsymbol{M}=mathbf{5 0}^{circ} ) If
( angle L M N sim triangle P Q R ) then the value of ( angle R )
is
A ( .40^{circ} )
B . ( 60^{circ} )
( c cdot 70^{circ} )
D. ( 110^{circ} )
10
152 n Fig. ( boldsymbol{A D} perp boldsymbol{C D} ) and ( boldsymbol{C B} perp boldsymbol{C D .} ) If
( A Q=B P ) and ( D P=C Q, ) prove that
( angle D A Q=angle C B P )
9
153 ( ln Delta P Q R, ) if ( P R^{2}=P Q^{2}+Q R^{2}, ) prove
that ( angle Q ) is right angle.
10
154 70. AABC is an isosceles triangle
where LB = 90°. If similar tri-
angles AABE and AACD are
formed on AB and AC then the
ratio of the areas of AABE and
AACD will be
(1) 1:2 (2) 1:3
(3) 2:3 (4) 3:4
9
155 In the adjoining figure ( boldsymbol{A B}= )
( 12 c m, C D=8 c m, B D=20 mathrm{cm} )
( angle A B D=angle A E C=angle E D C=90^{circ} . ) If
( B E=x, ) then
A. ( x ) has two possible values whose difference is 4
B. ( x ) has two possible values whose sum is 28
c. ( x ) has only one value and ( x geq 12 )
D. ( x ) cannot be determined with the given information
10
156 In ( triangle A B C, D ) is the midpoint of ( A B ) and ( E )
is mid point of AC. If area of ( triangle mathrm{ADE}=11 )
square units then area of ( triangle mathrm{ABC} ) is
A. 33 square units
B. 22 square units
c. 44 square units
D. None of the above
10
157 ( mathrm{PQ} ) 10
158 ( ln Delta L M N, angle L=60^{circ}, angle M=50^{circ} . ) If
( Delta L M N sim Delta P Q R, ) then the value of
( angle R ) is
A ( cdot 40^{circ} )
B. 30
( c cdot 70 )
D. ( 110^{circ} )
10
159 70. In a triangle, if three altitudes
are equal, then the triangle is
(1) Obtuse (2) Equilateral
(3) Right (4) Isosceles
9
160 In the given figure, ( boldsymbol{A} boldsymbol{E} | boldsymbol{D} boldsymbol{B}, boldsymbol{B C}= )
( mathbf{7} boldsymbol{c m}, boldsymbol{B} boldsymbol{D}=mathbf{5} boldsymbol{c m}, boldsymbol{D} boldsymbol{C}=mathbf{4} boldsymbol{c m} . ) If ( boldsymbol{C} boldsymbol{E}= )
( 12 c m, ) find ( A E ) and ( A C )
10
161 Fill in the blank.
Two circles are congruent if
9
162 In the figure above, ( A C=6 ) and ( B C= )
3. Point ( P(text { not shown }) ) lies on ( A B )
between ( A ) and ( B ) such that ( C P perp A B )
Which of the following could be the
length of ( C P ? )
( A cdot 2 )
B. 4
( c .5 )
D.
( E )
10
163 If the ratio of areas of two similar
triangle is 16: 81 find the ratio of their
corresponding sides.
10
164 In the figure, ( D E | B C )
(i) Prove that ( triangle A D E ) and ( triangle A B C ) are
( operatorname{similar} )
10
165 n Fig., ( D E F G ) is a square in a triangle
( A B C ) right angled at ( A )
Prove that
( Delta A G F sim Delta D B G )
ii) ( Delta A G F sim Delta E F C )
10
166 ngiven figure, ( frac{A O}{O C}=frac{B O}{O D}=frac{1}{2} ) and
( A B=5 mathrm{cm} . ) Find the value of DC.
10
167 In the given figure, ( A B | ) EF ( | ) CD. If ( A B= )
( 22.5 mathrm{cm}, mathrm{EP}=7.5 mathrm{cm}, mathrm{PC}=15 mathrm{cm} ) and DC
( =27 mathrm{cm} . ) Find ( mathrm{AC} )
( A cdot A C=37.5 mathrm{cm} )
B. AC = 22.5 cm
( c cdot A C=13.5 mathrm{cm} )
D. ( A C=27 mathrm{cm} )
10
168 Find the value of ( x ) when ( D E | A B )
A. 8
( B )
( c cdot 16 )
D. none
10
169 If ( D ) is a point on side ( A B ) of ( A B C ) and ( D E ) is a line through D meeting ( A C ) at ( E ) such that ( Delta A D E sim Delta A C B . ) Then ( A B )
AD is equal to
A . AE. AC
B. AC. DE
c. AE. BC
D. AB. BC
10
170 A man of height ( 1.8 m ) is standing near a Pyramid. If the shadow of the man is of length ( 2.7 m ) and the shadow of the
Pyramid is ( 210 m ) long at that instant, find the height of the Pyramid
10
171 In given figure ( P ) and ( Q ) are points on
sides ( A B ) and ( A C ) respectively of ( triangle A B C )
If ( A P=3 mathrm{cm}, P B=6 mathrm{cm}, A Q=5 mathrm{cm} ) and
( mathrm{QC}=10 mathrm{cm}, ) show that ( mathrm{BC}=3 mathrm{PQ} )
10
172 The areas of two similar triangles are ( 81 mathrm{cm}^{2} ) and ( 49 mathrm{cm}^{2} ) respectively. Find
the ratio of their corresponding heights. What is the ratio of their corresponding medians?
10
173 One side of a right-angles triangular scarf is ( 80 mathrm{cm} ) and its longest side is 1
( mathrm{m} . ) Find its cost at the rate of Rs 250 per
( boldsymbol{m}^{2} )
10
174 In the trapezium ( A B C D, ) side ( A B | ) side
DC. Diagonals ( A C ) and DB intersect each other at ( 0 . ) If ( A B=15, D C=10 ) find ( frac{O C}{O A} )
A ( cdot frac{O C}{O A}=frac{1}{3} )
в. ( frac{O C}{O A}=frac{7}{3} )
c. ( frac{O C}{O A}=frac{2}{3} )
D. ( frac{O C}{O A}=frac{5}{3} )
10
175 Show that ( triangle A B C, ) where
( boldsymbol{A}(-mathbf{2}, mathbf{0}), boldsymbol{B}(mathbf{2}, mathbf{0}), boldsymbol{C}(mathbf{0}, mathbf{2}) ) and ( triangle boldsymbol{P} boldsymbol{Q} boldsymbol{R} )
where ( boldsymbol{P}(-mathbf{4}, mathbf{0}), boldsymbol{Q}(mathbf{4}, mathbf{0}), boldsymbol{R}(mathbf{0}, mathbf{4}) ) are
similar.
10
176 In the figure given, if ( D E | B C ) and
( boldsymbol{A B}=mathbf{5} boldsymbol{x}-mathbf{4}, boldsymbol{B} boldsymbol{D}=mathbf{7} boldsymbol{x}-mathbf{5}, boldsymbol{C} boldsymbol{E}= )
( 5 x-3 ) and ( A D=3 x-2, ) find the value
of ( x )
4
B.
c. ( frac{7}{10} )
” ( frac{9}{10} )
10
177 ABCD a parallelogram. ( boldsymbol{E} ) is a point on ( A D ) and ( C E ) is produced to meet ( B A ) at
( F . ) If ( A E=4 c m, A F=8 mathrm{cm} ) and
( A B=12 mathrm{cm} . ) Find the perimeter (in ( mathrm{cm} )
of parallelogram ( boldsymbol{A B C D} )
10
178 In the given figure, the triangles are
congruent, Find the values of ( x ) and ( y )
9
179 The perimeters of two similar triangles
are ( 25 mathrm{cm} ) and ( 15 mathrm{cm} ) respectively. If one side of first triangle is ( 9 mathrm{cm}, ) then the corresponding side of the other triangle is
( A cdot 6.2 mathrm{cm} )
B. ( 3.4 mathrm{cm} )
c. ( 5.4 mathrm{cm} )
D. ( 8.4 mathrm{cm} )
10
180 If the corresponding sides of two triangles are proportional, then the two triangles are similar by which test
( A cdot ) SSS test
B. SAS test
C. AAA test
D. ASA test
10
181 We have seen how we can draw a series
of right triangles as in the picture.
What are the lengths of the sides the
tenth triangle?
10
182 67. The length (in metres) of the long-
est rod that can be put in a room
of dimensions 10 m x 10 mx
5 m is
(1) 15/3 (2) 15
(3) 102 (4) 573
9
183 If the corresponding angles of two triangles are equal then they are always
congruent. The given statement
A . Is always true
B. Is always false
c. can be true
D. can be determined
9
184 Use the following figure to find ( A E ), if
( B D=4.1 mathrm{cm} )
A . ( 4.1 mathrm{cm} )
B. ( 8.2 mathrm{cm} )
c. ( 16.4 mathrm{cm} )
D. Data insufficient
10
185 ( Delta A B C sim )
4. ( Delta Q P R )
8. ( Delta P Q R )
c. ( Delta R P Q )
D. ( Delta P R Q )
10
186 State whether the following statement is true or false.

If two squares have equal areas, they
are congruent.
A. True
B. False

9
187 If ( A B C ) and DEF are similar triangles
such that ( angle A=47^{circ} ) and ( angle B=83^{circ} )
then ( angle boldsymbol{F} ) is
( mathbf{A} cdot 60^{circ} )
B. ( 70^{circ} )
( c .50^{circ} )
D. ( 100^{circ} )
10
188 The perimeter of a triangle is equal to ( boldsymbol{K} )
times the sum of its altitude, then ( boldsymbol{K} ) is:
( mathbf{A} cdot mathbf{1} )
B. any number
c. less than 1
D. greater than 1
10
189 In the figure, seg ( P S ) intersects ( operatorname{seg} T K )
in the point ( R . angle T ) and ( angle K ) are right
angles. State whether ( triangle P T R ) and
( triangle S K R ) are similar. If yes, by which
test?
10
190 62. The area of the triangle, formed
by the graph of ax + by = 0
(where a, b are two positive real
numbers) and the co-ordinate
axes, is
(1)
sq. unit
sq. unit
sq. unit
sq. unit
9
191 How many trapezoids congruent to this
one:
are there in the following diagram?
( A cdot 8 )
B. 12
( c cdot 16 )
D. 2
9
192 State true or false:
With reference to the figure, ( R ) is mid-point of ( B C )
A. True
B. False
10
193 If in ( triangle P Q R ) and ( triangle L M N, angle P=angle M= )
( mathbf{6 0}^{circ}, boldsymbol{P Q}: boldsymbol{M} boldsymbol{L}=boldsymbol{P} boldsymbol{R}: boldsymbol{M} boldsymbol{N} ) and ( angle boldsymbol{N}= )
( 55^{circ}, ) then ( angle Q ) is:
A .50
B. 55
( c cdot 65 )
D. 75
10
194 Find the value of ( x ) of ( y ) using the information shown in the figure. Find
the measure of ( angle A B D ) and ( angle A C D )
9
195 In figure, ( P S ) is the bisector of ( angle Q P R ) of
( triangle P Q R ) Prove that
( frac{Q S}{S R}=frac{P Q}{P R} )
10
196 Two triangles are similar, if their corresponding angles of two triangles areequal and corresponding sides are in the same ratio. Is it true?
A. Yes
B. Not of same size
c. Not sure
D. cannot be possible
10
197 61. If the perimeter of a right-an-
gled isosceles triangle is
(4+2+4) cm, the length of
the hypotenuse is :
(1) 4 cm (2) 6 cm
(3) 8 cm (4) 10 cm
10
198 For two figures to be congruent, they need to be:
A. exactly alike
B. smaller
c. bigger
D. none of these
9
199 In the figure given below, ( boldsymbol{C D} | boldsymbol{E} boldsymbol{F} )
( A B . ) If ( A B=22.5 mathrm{cm}, E P=7.5 mathrm{cm} )
( P C=15 mathrm{cm} ) and ( D C=27 )
m. Calculate ( boldsymbol{E} boldsymbol{F} )
A. ( E F=14.5 mathrm{cm} )
3. ( E F=13.5 mathrm{cm} )
( c . E F=12.5 mathrm{cm} )
Done of the above
10
200 The corresponding sides of two similar triangles are in the ratio ( 1: 3 . ) If the area
of the smaller triangle in ( 40 mathrm{cm}^{2} ), find
the area of the larger triangle.
10
201 If the bisector of an angle of a triangle bisects the opposite side, then prove that the triangle is isosceles. 9
202 A line segment DE is drawn parallel to base BC of ( Delta A B C ) which cuts AB at
point ( mathrm{D} ) and ( mathrm{AC} ) at point ( mathrm{E} ). If ( mathrm{AB}=5 mathrm{BD} ) and ( mathrm{EC}=3.2 mathrm{cm} . ) Find the length of AE.
A . ( 12.8 mathrm{cm} )
в. ( 1.28 mathrm{cm} )
c. ( 2.8 mathrm{cm} )
D. 12.6 ( mathrm{cm} )
10
203 In triangle ( A B C, ) angle ( B ) is obtuse. ( D )
and ( E ) are mid-points of sides ( A B ) and
( B C ) respectively and ( F ) is a point on
side ( A C ) such that ( E F ) is parallel to ( A B )
Then, ( B E F D ) is a parallelogram. State
True or False.
A. True
B. False
10
204 State whether the following statement
is true or false.
All squares are congruent.
A. True
B. False
9
205 A man starts journey from
home. He goes 5 kms to the
North, then proceeds 10 kms to
the right. From there he again
turns right and goes 10 kms.
How far is he from home? All
distances are measured aerial-
ly.
(1) 200
(3) 225
(2) 150
(4) 1125
10
206 condition is not considered for
the similarity of triangle.
A. ( S A S )
в. ( S S S )
c. ( A A A )
D. ( A S A )
10
207 The sides of a triangle are ( 3 x+ )
( 4 y, 4 x+3 y ) and ( 5 x+5 y ) units, where
( boldsymbol{x}, boldsymbol{y}>0 . ) The triangle is
A. right angled
B. equilateral
c. obtuse angled
D. none of these
10
208 The areas of two similar triangles ABC
and PQR are ( 64 mathrm{cm}^{2} ) and ( 121 mathrm{cm}^{2} )
respectively. If ( Q R=15.4 mathrm{cm}, ) find ( B C )
10
209 How many triangles are there in the
following figure?
4.10
B . 24
( c cdot 22 )
D. 20
10
210 In given figure, ( D E | A C ) and ( D C | A P ) Prove that ( frac{boldsymbol{B} boldsymbol{E}}{boldsymbol{E C}}=frac{boldsymbol{B C}}{boldsymbol{C P}} ) 10
211 In the above figure, if DE ( | mathrm{BC} ), then ( x )
equals:
( A cdot 6 mathrm{cm} )
B. ( 7 mathrm{cm} )
( c cdot 3 mathrm{cm} )
D. ( 4 mathrm{cm} )
10
212 In the given figure, ( boldsymbol{A B} perp boldsymbol{B C}, boldsymbol{F G} perp )
( B C ) and ( D E perp A C . ) Prove that ( triangle A D E )
( sim triangle G C F )
10
213 Find ( mathrm{BC} ), if ( mathrm{AB}=7.2 mathrm{cm} )
A. ( 7.2 mathrm{cm} )
B. ( 7.1 mathrm{cm} )
( c cdot 0.72 mathrm{cm} )
D. ( 72 mathrm{cm} )
10
214 State True or False. If false, give reasons
for that
If two triangles are congruent, their corresponding angle are equal
A. True
B. False
9
215 In triangle ( A B C ; M ) is mid-point of ( A B, N ) mid-point of ( A C ) and ( D ) is any point in base BC. Then:
( A . ) MN bisects AD
B. MN divides AD in the ratio 1: 3
c. MN divides AD in the ratio 1: 2
D. MN divides AD in the ratio 1: 4
10
216 In right angled ( Delta A B C angle B= )
( 90^{circ} B D perp A C, A B=b, B D=c, B C= )
( boldsymbol{a}, boldsymbol{A} boldsymbol{D}=boldsymbol{8} boldsymbol{D} boldsymbol{C}=mathbf{1 0} . ) Then find ( ^{boldsymbol{}} boldsymbol{b}^{prime} )
A ( cdot sqrt{5} )
B. 12
( c cdot 6 sqrt{6} )
D. ( sqrt{18} )
10
217 ( 4 mathrm{CR}=mathrm{AB} )
If the above statement is true then
mention answer as 1 , else mention 0 if
false
10
218 In the given ( Delta A B C, ) if ( A B=A C ) and
( B D=D C, ) then ( angle A D C= )
( mathbf{A} cdot 60^{circ} )
B. ( 120^{circ} )
( mathbf{c} cdot 90^{circ} )
D. ( 45^{circ} )
10
219 Which of the following numbers form pythagorean triplet?
¡) 2,3,4
ii) 6,8,10
iii) 9,10,11
iv) 8,15,17
A . (ii), (iv)
B. (i), (ii)
c. (i), (ii), (iii)
D. (ii), (iii)
10
220 In a ( triangle A B C ) BD and ( mathrm{CE} ) are the altitudes
Prove that ( triangle A D B ) and ( triangle A E C ) are
similar. Check whether
( triangle C D B ) and ( triangle B E C ) are similar
10
221 64. In the adjoining figure if m ZABC
= m ZACE, then A ABC is :
D
(1) right-angled
(2) isosceles
(3) equilateral
(4) obtuse-angled
9
222 f ( triangle boldsymbol{P} boldsymbol{Q} boldsymbol{R} cong triangle boldsymbol{C} boldsymbol{A} boldsymbol{B}, boldsymbol{P} boldsymbol{Q}=boldsymbol{C} boldsymbol{A} . ) If true
enter 1 else 0
9
223 For a ( Delta A B C, A B=4 mathrm{mm}, B C=5 )
( mathrm{mm} ) and ( boldsymbol{A C}=mathbf{6} ) mm. If ( boldsymbol{Delta} boldsymbol{A} boldsymbol{B} boldsymbol{C} cong )
( Delta D E F, ) then ( E F=_{-}- )
( mathbf{A} cdot 4 mathrm{mm} )
B. ( 6 mathrm{mm} )
( c .5 mathrm{mm} )
D. cannot be determined
9
224 In the given figure you find two
triangles. Indicate whether the triangles
are similar. Give reasons in support of
your answer.
10
225 In figure, ( angle B A C=90^{circ} ) and ( A D perp B C )
Then
( mathbf{A} cdot B D cdot C D=B C^{2} )
B. ( A B . A C=B C^{2} )
c. ( B D . C D=A D^{2} )
D. ( A B . A C=A D^{2} )
10
226 Triangle ( X Y Z ) is right-angled at vertex
( Z . ) Calculate the length of ( Y Z, ) if ( X Y= )
( 13 mathrm{cm} ) and ( X Z=12 mathrm{cm} )
10
227 In the following figure, ( X Y ) is parallel to
( B C, A X=9 mathrm{cm} . X B=4.5 mathrm{cm} ) and
( B C=18 mathrm{cm} . ) Find ( X Y ) in cm.
10
228 In the figure ( : angle P S Q=90^{circ}, P Q= )
( mathbf{1 0} c boldsymbol{m}, boldsymbol{Q} boldsymbol{S}=boldsymbol{6} boldsymbol{c m} ) and ( boldsymbol{R} boldsymbol{Q}=boldsymbol{9} boldsymbol{c m} )
Calculate the length of ( boldsymbol{P} boldsymbol{R} )
10
229 The area of two similar triangles are in
ratio ( 16: 81 . ) Find the ratio of its sides.
A ( cdot frac{1}{9} )
в. ( frac{2}{9} )
( c cdot frac{3}{9} )
D. ( frac{4}{9} )
10
230 ( P ) and ( Q ) are the mid points on the sides
( C A ) and ( C B ) respectively of triangle
( A B C ) right angled at ( C . ) Prove that
( 4left(A Q^{2}+B P^{2}right)=5 A B^{2} )
10
231 Consider the following statements:
i) If three sides of a triangle are equal to three sides of another triangle, then the triangles are congruent.
ii) If the three angles of a triangle are equal to three angles of another triangle
respectively, then the two triangles are
congruent.
A . (i) is correct and (ii) is false
B. Both (i) and (ii) are false
c. Both (i) and (ii) are correct
D. (i) is false and (ii) is correct
9
232 If altitudes ( mathrm{CE} ) and ( mathrm{BD} ) of a triangle ( mathrm{ABC} )
are equal, then ( A B=dots )
A . AC
B. сE
( c . ) ВD
D. BC
10
233 The given figure shows a parallelogram
ABCD.E is a point in ( A D ) and ( C E )
produced meets BA produced at point ( F )
If ( A E=4 c m, A F=8 mathrm{cm} ) and ( A B=12 )
cm. find the perimeter of the parallelogram ABCD.
( A cdot 48 mathrm{cm} )
B. ( 44 mathrm{cm} )
( c .60 mathrm{cm} )
D. ( 50 mathrm{cm} )
E . ( 45 mathrm{cm} )
( F .54 mathrm{cm} )
G. none of the above
10
234 In the following figure,’ ( P^{prime} ) is point
equidistance from two lines ( ^{prime} l^{prime} ) and ( ^{prime} m^{prime} )
intersecting at point ( ^{prime} A^{prime} . ) Show that the
line ( A P ) bisect the angle between them.
9
235 n fig, ( A B C D ) is a parallelogram and
( B C ) is produced to a point ( Q ) such that
( boldsymbol{A D}=boldsymbol{C Q} . ) If ( boldsymbol{A Q} ) intersect ( boldsymbol{D} boldsymbol{C} ) at ( boldsymbol{P} )
show that ( a r(B P C)=a r(D P Q) )
9
236 n fig., ( C M ) and ( R N ) are respectively
then medians of ( triangle A B C ) and ( triangle P Q R )
prove that:
( triangle A M C sim triangle P N R )
10
237 In the given triangle, ( A B ) is parallel to
( boldsymbol{P Q} cdot boldsymbol{A P}=boldsymbol{c}, boldsymbol{P C}=boldsymbol{b}, boldsymbol{P Q}=boldsymbol{a}, boldsymbol{A B}=boldsymbol{x} )
What is the value of ( x ? )
A・ ( x=a-frac{a c}{b} )
B. ( x=b+frac{a c}{b} )
c. ( x=b-frac{a c}{b} )
D. ( x=a+frac{a c}{b} )
10
238 If both the rectangles are similar, find ( x )
( A )
B.
( c )
D. none of the above
10
239 n right triangle ( A B C ), right-angled at
( C, M ) is the mid-point of hypotenuse
( A B . C ) is joined to ( M ) and produced to a
point ( D ) such that ( D M=C M . ) Point ( D )
is joined to point ( B ).Which of the
following is correct.

This question has multiple correct options
A. ( Delta mathrm{AMC} cong Delta mathrm{BMD} )
B. ( angle D B C ) is a right angle.
c. ( Delta D B C equiv Delta A B C )
D. ( mathrm{CM}=frac{1}{2} mathrm{AB} )

9
240 Give two different examples of pair of
(i)
similar figures
10
241 The given figure shows a parallelogram
( A B C D ) with area ( 324 s q . c m . P ) is a
point in ( A B ) such that ( A P: P B=1: 2 )
Find:the ratio ( O P: O D )
4: 3
в. 2:
( c cdot 1: 4 )
D. 4: 3
10
242 73. In the given figures, the lengths
of the sides of AABC and APOR
are given and they are given in
same units. Also ZA and ZB are
given. Then value of ZP is
3.2
3.8
800
60°
60°
B6 C
7.6
P12
(1) 42°
(2) 36°
(3) 38°
(4) 40°
10
243 If ( angle D cong angle B ) and ( angle B cong angle Q, ) then ( angle D cong )
( angle Q ) is a ( —— ) property of
congruence.
A. reflexive
B. transitive
c. symmetric
D. addition postulate
9
244 Which of the following is a pythagorean triplet?
в. (5,7,9)
( mathbf{D} cdot(8,15,17) )
10
245 The diagonal BD of parallelogram ABCD
intersects ( A E ) at ‘F’.’ E’ is any point on BC.
Prove that DE.EF = FB. FA
10
246 f DE ( | ) BC and ( A D=1.7 c m, A B= )
( 6.8 c m ) and ( A C=9 c m . ) Then length of
AE is
A . ( 2.25 mathrm{cm} )
B. ( 4.5 mathrm{cm} )
( c .3 .4 mathrm{cm} )
D. ( 5.1 mathrm{cm} )
10
247 Which of the following is a Pythagorean triplet?
A. 3,4,5
в. 5,12,14
c. 6,8,11
D. 8,5,17
10
248 At a certain time of the day, a man 6
feet tall, casts his shadow 8 feet long. Find the length of the shadow cast by a
building 45 feet high, at the same time
which is next to the man
10
249 If two non-parallel lines are perpendicular to two other straight lines, each to each. Show that the acute
angle between the first pair of lines is
equal to the acute angle between the second pair of lines.
9
250 State true or false:
In quadrilateral ( A B C D, A D=B C ) and
the diagonals ( A C ) and ( B D ) intersect at
point ( boldsymbol{O}, ) such that ( frac{boldsymbol{O} boldsymbol{C}}{boldsymbol{O} boldsymbol{A}}=frac{boldsymbol{O} boldsymbol{D}}{boldsymbol{O B}}=frac{1}{mathbf{3}}, ) then the quadrilateral
( A B C D ) is a trapezium.
A. True
B. False
10
251 In ( triangle mathrm{PQR}, mathrm{M} ) and ( mathrm{N} ) are points on sides PQ and PR respectively such that PM = ( 15 mathrm{cm} ) and ( mathrm{NR}=8 mathrm{cm} . ) If ( mathrm{PQ}=25 mathrm{cm} ) and
( P R=20 mathrm{cm} ) state whether MN ( | mathrm{QR} )
10
252 ( A B C ) and ( D B C ) are two isosceles
triangles on the same base ( B C . ) Then
( angle A B D=angle A C D )
A. True
B. False
10
253 If two triangles are ( _{–}- ) they are similar.
A. Not equal
B. Equiangular
c. Different
D. Not proportionate
10
254 fthree or more parallel lines are intersected by two transversals prove
that the intercepts made by them on the
transversals are proportional
10
255 Find the ( x ) in terms of ( a, b, c ) 10
256 ( ln triangle A B C ) is ( angle B ) is right angle. If ( a=16 )
and ( c=12 ) then ( b= )
A . 8
B . 18
c. 20
D . 28
10
257 Fill in the blanks to make the
statements true,
In right triangle, the hypotenuse is the side
9
258 70. The circumcentre of a right-an-
gled triangle lies
(1) at the right angular vertex
(2) within the triangle
(3) outside the triangle
(4) on its hypotenuse
9
259 In a triangle ( A B C ), a straight line parallel to ( B C ) intersects ( A B ) and ( A C )
at point ( D ) and ( E ) respectively. If the
area of ( A D E ) is one-fifth of the area of
( A B C ) and ( B C=10 mathrm{cm}, ) then ( D E )
equals
A . ( 2 mathrm{cm} )
в. ( 2 sqrt{5} ) ст
( c .4 mathrm{cm} )
D. ( 4 sqrt{5} mathrm{cm} )
10
260 69. If the perimeter of a right-an-
gled triangle is 56 cm and area
of the triangle is 84 sq. cm,
then the length of the hypote-
nuse is (in cm)
(1) 25
(2) 50
(3) 7
(4) 24
9
261 n figure 2 , if ( D E | B C ), then the value
of ( x ) is equal to
( A cdot 3 mathrm{cm} )
B. ( 4 mathrm{cm} )
( c cdot 7 c m )
D. ( 4.7 mathrm{cm} )
10
262 n the given fig 0 is a point in the
nterior of a triangle of a triangle ( A B C, O D perp B C, O E perp A C ) and
( O F perp A B ) show that
( mathbf{O} mathbf{A}^{2}+mathbf{O B}^{2}+mathbf{O C}^{2}-mathbf{O D}^{2}-mathbf{O E}^{2}- )
( mathrm{OF}^{2}=mathrm{AF}^{2}+mathrm{BD}^{2}+mathrm{CE}^{2} )
10
263 n given figure ( D ) and ( E ) are respectively
the points on the sides ( A B ) and ( A C ) of a
( triangle A B C ) such that ( A B=5.6 mathrm{cm}, A D= )
1.4 ( c m, A C=7.2 mathrm{cm} ) and ( A E=1.8 mathrm{cm} )
show that ( D E | B C )
10
264 f ( triangle A B C sim triangle A D E ) and ar ( (triangle A D E)= )
( 9 operatorname{ar}(triangle A B C) ) then ( frac{B C}{D E} ) is equal to
( A )
B. 1
( c cdot frac{3}{4} )
D. None of these
10
265 ( ln triangle D E F, ) Line PQ ( | ) side EF. Find DP.
( boldsymbol{D} boldsymbol{Q}=mathbf{1 . 8} mathrm{cm}, boldsymbol{Q} boldsymbol{F}=mathbf{5} . boldsymbol{4} mathrm{cm}, boldsymbol{E} boldsymbol{P}=mathbf{7 . 2} )
( mathrm{cm} )
A ( .2 .4 mathrm{cm} )
B. ( 4.2 mathrm{cm} )
( c .3 .5 mathrm{cm} )
D. ( 1.8 mathrm{cm} )
10
266 ( triangle boldsymbol{A B C}, boldsymbol{X} boldsymbol{Y} | boldsymbol{B C}, frac{boldsymbol{A} boldsymbol{Y}}{boldsymbol{C Y}}=frac{1}{2} ) and
( A X=4 . ) Find ( B X )
10
267 Line segment ( A B ) is parallel to another
ine-segment ( C D, O ) is the midpoint of
AD.Show that ( triangle A O B cong triangle D O C )
9
268 Choose the correct criteria to prove that two triangles are congruent:
Criteria:
A) All three corresponding sides are
congruent.
B) Two angles and the side between
them are congruent.
C) Two angles and a non-included side
are congruent.
D) Two sides and the angle between
them are congruent. For the given congruency theorem which option is suitable.
1) SSS (side-side-side) =
2) SAS (side-angle-side) =
3) AAS (angle-angle-side) =
4) ASA (angle-side-angle) ) =
9
269 In the figure, the line segment ( X Y ) is
parallel to the side ( A C ) of ( triangle A B C ) and it
divides the triangle into two parts of equal areas, then match the column.
10
270 In the figure, ( A P=3 mathrm{cm}, A R= )
( 4.5 mathrm{cm}, A Q=6 mathrm{cm}, A B=5 mathrm{cm}, ) and
( A C=10 mathrm{cm} . ) Find the length (in ( mathrm{cm} ) ) of
( boldsymbol{A D} )
10
271 State true or false:
In parallelogram ( A B C D . E ) is the mid-
point of ( A B ) and ( A P ) is parallel to ( E C )
which meets ( D C ) at point ( O ) and ( B C )
produced at ( boldsymbol{P} ). Hence
( O ) is mid-point of ( boldsymbol{A} boldsymbol{P} )
A. True
B. False
10
272 An insect ( 8 m ) away from the foot of a lamp post which is ( 6 m ) tall, crawls towards it. After moving through a distance, its distance from the top of the lamp post is equal to the distance it has moved. How far is the insect away
from the foot of the lamp post? [Bhaskaracharya’s Leelavathi]
10
273 70. If one of the equal angles of an
isosceles triangle is 65°, then the
angle at the vertex is
(1) 70° (2) 50°
(3) 60°
(4) 40°
9
274 61.
The point of intersection of the
altitudes of a triangle is known
as
(1) Centroid
(2) In-centre
(3) Orthocentre
(4) Circumcentre
9
275 Show that ( left(boldsymbol{m}^{2}-mathbf{1}right),(boldsymbol{2 m}), boldsymbol{m}^{2}+mathbf{1} )
always form a pythagoran triplet.
10
276 STATEMENT – 1: If tangents OR, PR, PQ and drawn respectively at ( A, B, C ) to the circle circumscribing an acute-angled
( Delta A B C ) so as the form another ( Delta P Q R ) then the ( angle R P Q=angle B A C )
STATEMENT – 2: ( Delta A B C ) is similar to ( Delta )
CPB.
A. Statement – 1 is True, Statement – 2 is True, Statement 2 is a correct explanation for Statement – 1
B. Statement-1 is True, Statement – 2 is True : Statement 2 is NOT a correct explanation for Statement- –
c. Statement – 1 is True, Statement- – 2 is False
D. Statement-1 is False, Statement – 2 is True
10
277 71. The heights of two similar right-
angled triangles A LMN and A
OPG are 48 cm and 36 cm. If
OP = 12 cm, then LM is
(1) 10/6
1
3
cm
(2) 16 cm
(3) 20 cm
(4) 12 cm
10
278 67. In two similar triangles ABC and
MNP, if AB = 2.25 cm, MP = 4.5
cm and PN = 7.5 cm and m
ZACB = m ZMNP and m ZABC
= m ZMPN, then the length of
side BC, in cm, is
(1) 4.5 (2) 3.75
(3) 4.75 (4) 3.5
10
279 ( P ) is a point on side ( B C ) of ( a ) parallelogram ABCD. If DP produced meets ( A B ) produced at point ( L ), then ( D L: D P=A L: B C )
If the above statement is true then
mention answer as 1 , else mention 0 if
false
10
280 55. On decreasing each side of an
equilateral triangle by 2 cm,
there is a decrease of 4 13 cm2
in its area. The length of each
side of the triangle is
(1) 8 cm (2) 3 cm
(3) 5 cm (4) 6 cm
10
281 Two triangles are ……….. if two angles and included side (common to both the
angles) are equal to two angles and included side (common to both angles) of the other triangle.
A. unequal
B. congruent
c. equilateral
D. none of these
9
282 In given figure, If PSIIQR, prove that
( triangle P O S sim triangle R O Q )
10
283 n ( triangle A B C, angle A B C=90^{circ}, A D= )
( D C, A B=12 mathrm{cm} ) and ( B C=6.5 mathrm{cm} )
Find the area of ( triangle A D B )
10
284 If ( Delta A B C cong Delta D E F, angle A=47^{circ}, angle E= )
( 83^{circ}, ) then the value of ( angle C ) is:
A ( cdot 47^{circ} )
B. 30
c. 40
( D cdot 50^{circ} )
9
285 “If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent”. Is the statementt true? Why? 9
286 If ( tan theta=frac{3}{4} ) Find ( 3 cos A+4 sin A ) 10
287 In two right triangles one side and an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are
congruent.
9
288 If ( boldsymbol{L} boldsymbol{M} | boldsymbol{Q} boldsymbol{R}, frac{boldsymbol{P} boldsymbol{L}}{boldsymbol{L} boldsymbol{Q}}=frac{boldsymbol{3}}{mathbf{5}}, boldsymbol{P} boldsymbol{R}=mathbf{5 . 6} boldsymbol{c m} )
then ( P M ) is
A ( .2 .4 mathrm{cm} )
B. 2.3 ст
( c .2 .1 mathrm{cm} )
D. 2.2 cm
10
289 In the triangle ( A B C, P Q ) is parallel to
( B C . ) Find the length of ( x ) and ( y )
A. ( x=26.25, y=17.1 )
B. ( x=17.1, y=26.25 )
c. ( x=17.34, y=26.57 )
D. ( x=26.45, y=17.33 )
10
290 In triangle ( P Q R L P=110^{circ} ) and ( L R= )
( 60^{circ} ) which side of the triangle is
smallest.
10
291 ( A C^{2}=B C times A B )
If the above statement is true then
mention answer as 1 , else mention 0 if
false
10
292 If the ratio of the corresponding sides of two similar triangles is 2: 3 then the ratio of their corresponding altitude is
A. 3: 5
B. 16: 81
( c cdot 4: 9 )
D. 2:3
10
293 In the following figure, ABCD is a
parallelogram.If ( angle A B P=x angle C B P ),then
find the value of ( boldsymbol{x} )
10
294 n given figure, ( D ) is a point on the side ( B C ) of ( triangle A B C ) such that ( angle B A C= )
( angle A D C ) prove that
( boldsymbol{C A}^{2}=boldsymbol{C B} times boldsymbol{C D} )
10
295 In the given figure ( Delta A B C cong Delta A B T )
write all the corresponding sides.
9
296 66. ABCD is a rhombus. A straight
line through C cuts AD produced
at P and AB produced at 9. If
DP =
AB, then the ratio of the
lengths of B9 and AB is
(1) 2:1 (2) 1:2
(3) 1:1 (4) 3:1
10
297 In triangle ( A B C ) is right-angled at ( B )
and ( B D ) is perpendicular to ( A C, ) then
find:
( cot angle D B A ) is ( frac{m}{12} . ) value of ( mathrm{m} ) is,
10
298 Find the value of ( c ) in the triangle using
Pythagoras theorem.
A .24
B. 25
( c .26 )
0.2
10
299 n the given figure ( A B C ) is a triangle. If ( frac{A D}{A B}=frac{A E}{A C}, ) then prove that DE II BC 10
300 State true or false:
In an acute-angled ( triangle A B C, A D ) is
perpendicular to side ( B C, ) then ( A C^{2}= )
( A B^{2}+B C^{2}-2 B C times B D )
A. True
B. False
10
301 triangles become always similar.
A. Acute-angled
B. Equilateral
c. obtuse-angled
D. None of the above
10
302 ( S ) and ( T ) are points on sides ( P R ) and ( Q R )
of ( triangle P Q R ) such that ( angle P=angle R T S . ) Show
that ( triangle boldsymbol{R} boldsymbol{P} boldsymbol{Q} sim triangle boldsymbol{R} boldsymbol{T} boldsymbol{S} )
10
303 ( mathrm{DE}=2.4 mathrm{cm}, ) find the length of BC 10
304 How many rectangles congruent to this rectangle are there in the following diagram?
( A cdot 6 )
B. 7
( c cdot s )
D. 15
9
305 Find the value of ( x ) and ( y ). (Use
Pythagoras theorem)
A ( . x=12 ) and ( y=3 )
3. ( x=10 ) and ( y=5 )
c. ( x=12 ) and ( y=5 )
( x=11 ) and ( y=5 )
10
306 17. Triangle is
a. acute angled
b. right angled but not isosceles
c. isosceles
d. isosceles right angled
9
307 A line parallel to the base of a triangle cuts the triangle into two regions of equal area. This line also cuts the altitude into two parts. Find the ratio of the two parts of the altitude.
A . 1: 1
B. 1: 2
c. ( 1: sqrt{2} )
D. ( 1:(sqrt{2}+1) )
10
308 ABCD is a quadrilateral; ( P, Q, R ) and ( S ) are
the points of trisection of sides
( A B, B C, C D ) and DA respectively and are adjacent to A and C; prove thar PQRS is a parallelogram.
10
309 The sides of right angles triangle are 63,9 find its hypotenuse. 10
310 In the figure below, the diagonal ( boldsymbol{A C} ) of
quadrilateral ( A B C D ) bisects ( angle B A D )
and ( angle B C D . ) Then ( B C=m D C ).Find ( m )
10
311 n the following figure, ( X Y ) is parallel to ( B C, A X=9 mathrm{cm} . X B=4.5 mathrm{cm} ) and
( B C=18 mathrm{cm} )
What is the value of ( frac{A Y}{Y C} ? )
10
312 In a triangle ( A B C, ) a line ( P Q ) is drawn
parallel to ( B C ), points ( P ), Q being on ( A B )
and ( A C ) respectively. If ( A B=3 A P )
then what is the ratio of the area of
triangle ( A P Q ) to the area of triangle
( A B C ? )
A .1: 3
B. 1: 5
c. 1: 7
D. 1: 9
10
313 In the figure if ( D E | B C ) and ( A D= )
( mathbf{3} boldsymbol{x}-mathbf{2}, boldsymbol{A} boldsymbol{E}=mathbf{5} boldsymbol{x}-mathbf{4}, boldsymbol{B} boldsymbol{D}=mathbf{7} boldsymbol{X}-mathbf{4} )
( B D-7 x-5 ) AND ( C E=5 x-3 . ) Find
( mathbf{x} )
10
314 In the following triplet pythagorean? show working (18,79,82) 10
315 1.
The sides of a triangle are 3x+4y, 4x+3y and 5x+5y wherex,
y>0 then the triangle is
[2002]
(a) right angled
(b) obtuse angled
© equilateral
(d) none of these
10
316 ( P M=4 mathrm{cm} ; Q M=4.5 mathrm{cm} )
( P N=4 mathrm{cm} ; mathrm{NR}=4.5 mathrm{cm} )
10
317 see how much time he would save
taking a shortcut to home from football
practice. He usually walked 6 blocks
south and 9 blocks east. Which picture
shows his shortcut?
( A )
B.
( c )
D.
10
318 In the given figure, ( D E | B C ) and
( A D: D B=5: 4, ) find ( frac{operatorname{area}(Delta D F E)}{operatorname{area}(Delta C F B)} )
4. 5: 9
B . 25: 16
c. 25: 81
D. 81: 25
10
319 In ( Delta A B C, angle A=90^{circ} ) and ( A D perp B C )
Then, ( boldsymbol{A} boldsymbol{D}^{3}=boldsymbol{B} boldsymbol{D} times boldsymbol{D} boldsymbol{C} )
A. True
B. False
10
320 us-
54. It is given that AABC AFDE in
which AB = 5cm, ZB= 40°, ZA=
80° and FD = 5cm. Then, which
of the following is true?
(1) ZD = 60°
(2) ZE= 60°
(3) ZF = 60°
(4) ZD = 80°
9
321 ( mathrm{DE}=mathrm{FE} ) 10
322 62. The circumcentre and the ortho-
centre of a triangle coincide. Then
(1) the centroid also coincides
with them
(2) the centroid will be different
from them
(3) the triangle is isosceles
(4) the triangle is right angled
9
323 61. The perimeters of two similar tri-
angles are 30 cm and 20 cm re-
spectively. If one side of the first
triangle is 9 cm. Determine the
corresponding side of the second
triangle
(1) 13.5 cm (2) 6 cm
(3) 15 cm (4) 5 cm
10
324 If the area of two similar triangles are equal, then they are
A . equilateral
B. isosceles
c. congruent
D. not congruent
10
325 When one acute angle of a triangle is equal to one acute angle of other triangle, and the triangles are right angles, do you think the triangles are similar?
A. Not sure
B. Similar
c. Not similar
D. cannot be possible
10
326 If ( triangle A B C cong triangle R Q P, angle A=80^{circ}, angle B= )
( 60^{circ}, ) then the value of ( angle P ) is
A . ( 60^{circ} )
В. ( 50^{circ} )
c. ( 40^{circ} )
D. ( 80^{circ} )
10
327 Given line ( A B ) is parallel to line ( C D )
( angle C H G ) and ( angle E G B ) are:
A. Non-congruent
B. Congruent
c. Complementary
D. None of these
9
328 63. In the figure A ACB – A APQ. If
BC = 8 cm, PQ4 cm, AP 2.8
cm, find CA:
(1) 8 cm (2) 6.5 cm
(3) 5.6 cm
(4) None of these
10
329 In triangle ( A B C, angle B=90^{circ} ) and ( D ) is
the mid-point of side Be. Prove that:
( A C^{2}=A D^{2}+3 C D^{2} )
10
330 ( operatorname{In} ) a trapezium ( boldsymbol{A B C D}, boldsymbol{A B} | boldsymbol{D C} ) and
( D C=2 A B . E F ) drawn parallel to ( A B )
cuts ( A D ) in ( F ) and ( B C ) in ( E ) such that ( frac{B E}{E C}=frac{3}{4} . ) Diagonal ( D B ) intersects ( E F )
at ( G . ) Prove that ( mathbf{7 F E}=mathbf{1 0 A B} )
10
331 ?
(4) 7.5
at 55. In A POR, PS is the bisector of
ZP and PT I QR, then TPS is
equal to :
TS
is
(1)29+ ZR 2)909 + Ž 40
(3) 90° -5R (4) (29-ZR)
10
332 68. In A ABC, AD is the internal bi-
sector of ZA, meeting the side BC
at D. If BD = 5 cm,
BC = 7.5 cm, then AB : AC is
(1) 2:1 (2) 1:2
(3) 4 : 5
(4) 3:5
10
333 In the given figure, apply ( operatorname{sss} ) congruence condition and state the
result in the symbolic form.
10
334 ( D ) is any point on side ( A C ) of a ( Delta A B C ) with ( A B=A C ) Show that ( C D<B D ) 10
335 In a right angled triangle, if the square of the hypotenuse is twice the product of the other two sides, then one of the
angles of the triangle is:
A ( cdot 15^{circ} )
B . ( 30^{circ} )
( c cdot 45^{circ} )
D. ( 60^{circ} )
10
336 In the figure below. PQR is a right-
angle triangle right angled at ( Q . X Y ) is
parallel to ( boldsymbol{Q} boldsymbol{R} . boldsymbol{P} boldsymbol{Q}=boldsymbol{6} mathrm{cm}, boldsymbol{P Y}=mathbf{4} mathrm{cm} )
and ( P X: X Q=1: 2 . ) Calculate the
lengths of ( boldsymbol{P R} ) and ( boldsymbol{Q} boldsymbol{R} )
A. ( P R=12 mathrm{cm} ; Q R=10.392 mathrm{cm} )
B. ( P R=13 mathrm{cm} ; Q R=11.392 mathrm{cm} )
c. ( P R=11 mathrm{cm} ; Q R=12.392 mathrm{cm} )
D. none of the above
10
337 ( mathrm{BP}=2 mathrm{AC} )
f the above statement is true then
mention answer as 1 , else mention 0 if
false
10
338 ( triangle A B C, D E | B C, ) find the value of ( x ) 10
339 If an equilateral triangle, having centroid at the origin, has a side along
the line, ( x+y=2 ), then the area (in
sq.units) of this triangle is:
( mathbf{A} cdot mathbf{6} )
B. ( 6 sqrt{3} )
c. ( frac{9}{2} sqrt{3} )
D. ( 3 sqrt{6} )
9
340 69. O is the orthocentre of AABC. If
ZBOC = 110°, ZBAC is equal to
(1) 70° . (2) 80°
(3) 110
(4) 90°
9
341 86. The length of each side of an equi-
lateral triangle is 1473 cm. The
area of the incircle, in cm, is
(1) 450 (2) 308
(3) 154 (4) 77
9
342 In a ( triangle A B C, B C=A B ) and ( angle B=80^{0} )
Then ( angle A ) is equal to?
A ( cdot 80^{circ} )
( ^{0} 0 cdot 8 cdot 8^{00} )
B . ( 40^{circ} )
( c cdot 50^{0} )
D. ( 100^{circ} )
10
343 In ( triangle A B C, ) if ( A D perp B C ) and ( A D^{2}= )
( B D times D C, ) prove that ( angle B A C=90^{0} )
10
344 n given figure, ( angle B A C=90^{circ} ) and
segment ( A D perp B C . ) Prove that ( A D^{2}= )
( boldsymbol{B} boldsymbol{D} times boldsymbol{D} boldsymbol{C} )
10
345 Anna went to the market to buy some
boxes to store things. She
was surprised to find boxes one inside
the other. They were ( ldots ldots ) boxes.
A. Not similar
B. Ambiguous
c. same size
D. similar
10
346 f ( triangle boldsymbol{D} boldsymbol{E} boldsymbol{F} cong triangle boldsymbol{R} boldsymbol{P} Q, ) then ( angle boldsymbol{D}=angle boldsymbol{Q} . ) If
true enter 1 else 0
9
347 Let ( X ) be any point on the side ( B C ) of ( a ) triangle ABC. If ( mathrm{XM}, mathrm{XN} ) are drawn parallel to BA and CA meeting CA, BA in M, N respectively; MN meets BC
produced in ( T, ) prove that ( boldsymbol{T} boldsymbol{X}^{2}=boldsymbol{T} boldsymbol{B} times )
( boldsymbol{T} boldsymbol{C} )
10
348 In ( triangle A B C, X Y | B C ) and ( X Y ) divides the triangle into two parts of equal areas. Find ( left(frac{B X}{A B}right) ) 10
349 Goldfish are sold at Rs.15 each. The
rectangular coordinate graph showing the cost of 1 to 12 goldfish is:
A. a straight line segment
B. a set of horizontal parallel line segments
c. a set of vertical parallel line segments
D. a finite set of distinct points
E a straight line
10
350 ( A B C D ) is a trapezium in which ( A B | ) ( D C, D C=7 c m ) distance between ( A B )
and ( D C ) is ( 4 c m ). Find ( A B ).
10
351 Check if the triangles are similar. If
similar, write the similarity in symbolic
form. Mention the similarity condition
used
( mathbf{A} cdot A S A )
B. ( S A S )
( c . A A S )
D. NONE
10
352 Under which congruence condition the following figure are said to be congruent:
(a) Two Line Segments
(b) Two squares
(c)Two rectangles
( (d) ) Two circles
9
353 Solve the following:
n ( Delta M N P, angle M N P= )
( 90^{circ}, operatorname{seg} N Q perp operatorname{seg} M P . ) If ( M Q= )
( mathbf{9}, boldsymbol{Q} boldsymbol{P}=mathbf{4}, ) then find ( boldsymbol{N} boldsymbol{Q} )
10
354 In given figure the diagonal BD of a parallelogram ABCD intersects the
segment AE at the point ( F ), where E is
any point on the side BC. Prove that
( boldsymbol{D} boldsymbol{F} times boldsymbol{E} boldsymbol{F}=boldsymbol{F} boldsymbol{B} times boldsymbol{F} boldsymbol{A} )
10
355 In given figures sides ( A B ) and ( A C ) and
median ( A D ) of a triangle ( A B C ) are
respectively proportional to sides ( boldsymbol{P Q} )
and ( P R ) and median ( P M ) of another
triangle ( P Q R . ) Show that ( triangle A B C sim )
( triangle boldsymbol{P} boldsymbol{Q} boldsymbol{R} )
10
356 In triangle ( A B C, ) medians ( A D ) and ( B E ) are drawn.lf ( boldsymbol{A D}=mathbf{4}, angle boldsymbol{D A B}=frac{boldsymbol{pi}}{mathbf{6}} ) and
( angle A B E=frac{pi}{3}, ) then the area of the
( triangle A B C ) is
A ( cdot frac{8}{3} )
в. ( frac{16}{3} )
c. ( frac{32}{3 sqrt{3}} )
D. ( frac{64}{3} )
10
357 n the figure (1) given below, AB I CR and LM | QR Prove that ( frac{boldsymbol{B} boldsymbol{M}}{boldsymbol{M} boldsymbol{C}}=frac{boldsymbol{A} boldsymbol{L}}{boldsymbol{L} boldsymbol{Q}} ) 10
358 If in triangle ( boldsymbol{X} boldsymbol{Y} boldsymbol{Z}, boldsymbol{X} boldsymbol{Y}=boldsymbol{X} boldsymbol{Z} ) and
( M, N ) are the mid points of ( X Y, Y Z ) then which one of the following is correct?
( mathbf{A} cdot M N=Y Z )
в. ( N Y=N Z=M N )
c. ( M X=M Y=N Y )
D. ( M N=M X=M Y )
10
359 n given figure, If ( boldsymbol{E} boldsymbol{F}|boldsymbol{D} boldsymbol{C}| boldsymbol{A} boldsymbol{B} ). prove
that ( frac{boldsymbol{A} boldsymbol{E}}{boldsymbol{E} boldsymbol{D}}=frac{boldsymbol{B} boldsymbol{F}}{boldsymbol{F} boldsymbol{C}} )
10
360 State whether the following statement is true or false.

If two rectangles have equal area, they
are congruent.
A. True
B. False

9
361 Two triangles are congruent if they have
the same and
A. Not sure
B. Shape
c. Both B and D
D. size
9
362 All congruent figures are similar but the similar figures are not congruent.Is this statement true or false?
( A ). False
B. Both A and C
c. True
D. Not applicable
10
363 n the given figure, ( angle B=60^{circ}, A B=8 )
( mathrm{cm} ) and ( mathrm{BC}=25 mathrm{cm} . ) Calculate :
(i) ( B E )
(ii) ( boldsymbol{A C} )
10
364 The perimeter of two similar triangles
( triangle A B C ) and ( triangle D E F ) are ( 36 mathrm{cm} ) and 24
( mathrm{cm} ) respectively. If ( D E=10 mathrm{cm}, ) then
( A B ) is :
A . ( 12 mathrm{cm} )
B. ( 20 mathrm{cm} )
c. ( 15 mathrm{cm} )
D. ( 18 mathrm{cm} )
10
365 In the given figure you find two
triangles. Indicate whether the triangles
are similar. Give reasons in support of
your answer.
10
366 In the adjoining figure, find the
measure of ( angle B A C, ) if ( angle A B D=angle C A D )
and ( angle B A D=angle A C D )
A ( cdot 120^{circ} )
В. ( 60^{circ} )
( c cdot 75^{circ} )
D. ( 90^{circ} )
10
367 Prove that if the lengths of two altitudes of a triangle are equal, then the triangle is isosceles. 10
368 Which of the following postulate can be used to prove ( Delta K L M ) and ( Delta R S T ) are similar?
This question has multiple correct options
A . AAS
B. sss
c. sas
D. AAA
10
369 Find ( x ) 10
370 A pole of 6 metres in height casts a shadow of 3.6 metres at a certain time
of the day. Find the length of the shadow
cast by a 4.5 metres tower at the same time.
10
371 In the figure ( angle A=angle C E D, C D= )
( mathbf{8} boldsymbol{c m}, boldsymbol{C} boldsymbol{E}=mathbf{1 0} boldsymbol{c m}, boldsymbol{B} boldsymbol{E}= )
( 2 c m, A B=9 c m, A D= )
( b ) and ( D E=a ). The value of ( a+b ) is
A. ( 13 mathrm{cm} )
В. ( 15 mathrm{cm} )
( c cdot 12 mathrm{cm} )
D. ( 9 mathrm{cm} )
10
372 State true or false:
With reference to the figure,
( mathbf{2} boldsymbol{P} boldsymbol{R}=boldsymbol{A} boldsymbol{B}+boldsymbol{A} boldsymbol{C} )
A. True
B. False
10
373 In figure, DE||BC, then the value of ( x )
equals to:
A. ( 2.5 mathrm{cm} )
B. ( 2 mathrm{cm} )
c. ( 1.4 mathrm{cm} )
D. ( 4 mathrm{cm} )
10
374 Fig. ( 6.36, frac{Q R}{Q S}=frac{Q T}{P R} ) and ( angle 1=angle 2 )
Show that ( Delta P Q S sim Delta T Q R )
10
375 STATEMENT – 1: If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.

STATEMENT – 2 : If in two triangles,
corresponding angles are equal, then their corresponding sides are in the
same ratio and hence the two triangles
are similar.
A. Statement – 1 is True, Statement – 2 is True, Statement 2 is a correct explanation for Statement –
B. Statement – 1 is True, Statement – 2 is True : Statement 2 is NOT a correct explanation for Statement –
c. Statement – 1 is True, Statement – 2 is False
D. Statement – 1 is False, Statement – 2 is True

10
376 If three sides of a triangle are respectively equal to three sides of
another triangle, then the triangles are:
A. unequal
B. equilateral
c. congruent
D. none of these
9
377 Rhombus PQRB is inscribed in ( triangle boldsymbol{A B C} )
such that ( angle B ) is one of the its angle, ( P, Q )
and ( mathrm{R} ) lie on ( mathrm{AB}, mathrm{AC} ) and ( mathrm{BC} ) respectively. If ( A B=12 mathrm{cm} ) and ( B C=6 mathrm{cm} ) find the
sides of rhombus PQRB.
10
378 65. The ratio of circumradius and
hypotenuse of a right-angled tri-
angle is
(1) 2:3 (2) 1:2
(3) 2:1 (4) 1:1
9
379 In ( triangle boldsymbol{A B C} ) and ( triangle boldsymbol{D E F}, angle boldsymbol{B}=angle boldsymbol{E}= )
( 90^{circ} ; A C=D F a n d B C=E F )
Then
triangles are congruent.
A. True
B. False
9
380 In given figure ( A B C ) is a triangle in
which ( A B=A C ) and ( D ) is a point on ( A C )
such that ( B C^{2}=A C times C D . ) Prove that
( mathrm{BD}=mathrm{BC} )
10
381 53. In the given figure, ZA = 80°, ZB
= 60°, ZC = 2x and ZBDC = yº.
BD and CD bisect angles B and
C respectively. The values of x
and y respectively are :
BE
(1) 15° and 70°
(2) 10° and 160°
(3) 20° and 130°
(4) 20° and 125°
10
382 n ( triangle A B C ) and ( triangle P Q R, ) if ( A B= ) ( 2.5 c m, A C=3.5 c m, B C= )
( 4.0 c m, P Q=5 c m, P R=7 c m ) and
( Q R=8 c m, ) then find whether ( triangle A B C )
and ( triangle P Q R ) are similar or not
10
383 For two acute angled ( triangle A B C ) and ( triangle P Q R ) if ( triangle A B C sim triangle P Q R ) then prove
( operatorname{that} frac{operatorname{area}(triangle A B C)}{operatorname{area}(triangle P Q R)}=frac{A B^{2}}{P Q^{2}}=frac{B C^{2}}{Q R^{2}}= )
( frac{A C^{2}}{P R^{2}} )
10
384 In the adjoining figure, D is a point on
the side BC of ( Delta A B C ) such that
( angle A D C=angle B A C ) Then ( frac{C A}{C D}=frac{C B}{C A} )
A. True
B. False
10
385 In a circle with centre ( boldsymbol{O}, boldsymbol{O} boldsymbol{D} perp ) chord
( A B . ) If ( B C ) is the diameter, then which
of the following option is correct:
( mathbf{A} cdot A C=B C )
В ( . O D=B D )
c. ( A C=2 times O D )
D. None of these
10
386 If the areas of two similar triangles are equal, then they are
A . equilateral
B. isosceles
c. congruent
D. not congruent
9
387 PQR is a triangle right angled at ( P ) and M is a point on QR such that PM ( perp ) QR. Show that ( boldsymbol{P} boldsymbol{M}^{2}=boldsymbol{Q} boldsymbol{M} cdot boldsymbol{M} boldsymbol{R} ) 10
388 Corresponding sides of two similar triangles are in the ratio ( 2: 3 . ) If the area
of the smaller triangle is ( 48 mathrm{cm}^{2} ) determine the area of the larger triangle.
10
389 If the area of two similar triangles are equal, prove that they are congruent. 9
390 In given figure, DEFG is a square and ( angle B A C=90^{0} . ) Prove that
(i) ( triangle A G F sim triangle D B G )
(ii) ( triangle A G F sim triangle E F C )
(iii) ( triangle D B G sim triangle E F C )
( (text { iv }) boldsymbol{D} boldsymbol{E}^{2}=boldsymbol{B} boldsymbol{D} times boldsymbol{E} boldsymbol{C} )
10
391 65. AABC is an isoscles triangle with
AB = AC and ZA = x, if side BA
is produced such that AB = AD
then what will be the value of
ZBCD?
(1) 90° – X (2) 90° + x
(3 90° (4) 2x
9
392 Two angles of triangle ( A B C ) are ( 85^{circ} ) and
( 65^{circ} ) while the two angles of another
triangle DEF are ( 30^{circ} ) and ( 65^{circ} . ) Which of the statements is correct?
A. ( triangle ) ABC is similar to ( triangle ) DEF
B. ( triangle ) ABC is congurent to ( triangle ) DEF
c. ( triangle ) ABC is equal to ( triangle ) DEF
D. None of these
10
393 ( A B C ) and ( B D E ) are two equilateral
triangles such that ( D ) is the midpoint of
( B C . ) Ratio of the areas of triangles ( A B C )
and ( B D E ) is
A .2: 1
B. 1: 2
( c cdot 4: 1 )
D. 1: 4
10
394 n given figure, If ( triangle P O S sim triangle R O Q )
prove thar PSI|QR.
10
395 60. In AABC, it is given that D is the
midpoint of BC; E is the mid-
point of BDand O is the midpoint
of AE.Then, ar(ABOE) = ?
B
E
D
(1) ar(ABC)
(2) artAABC
(3) ar(ABC
14) ar(ABC
9
396 Find DF, if ( mathrm{CG}=11 mathrm{cm} )
A. ( 5.6 mathrm{cm} )
B. ( 5.5 mathrm{cm} )
( c cdot 0.55 mathrm{cm} )
D. ( 55 mathrm{cm} )
10
397 58. I is the incentre of a triang
ABC. If ZABC = 65° and ZACH
= 55°. then the value of BIC
is
(1) 130°
(3) 140°
(2) 120°
(4) 110
9
398 If ( Delta A B C cong Delta X Y Z, ) which of the
following statements is incorrect?
( mathbf{A} cdot angle A=angle X )
В. ( angle B=angle Z )
c. ( A B=X Y )
( mathbf{D} cdot B C=Y Z )
9
399 ( A B C D ) is a rectangle inscribed in a
quadrant of a circle having radius
10cm. If ( A D=2 sqrt{5} c m ), find the area of
( A B C D )
( A cdot 30 c m^{2} )
3. ( 50 mathrm{cm}^{2} )
( c cdot 40 c m^{2} )
D. ( 35 c m^{2} )
10
400 In an equilateral triangle if 3 times the square of one side is equal to ( mathrm{K} ) times the square of its altitude then Kequals
A ( cdot frac{4}{3} )
B. 2
( c cdot 4 )
D. ( frac{9}{4} )
10
401 70. The lengths of three medians
of a triangle are 9 cm, 12 cm
and 15 cm. The area (in sq. cm)
of the triangle is
(1) 24
(2) 72
(3) 48
(4) 144
10
402 n given figure, AD and CE are two altitudes of ( triangle A B C ). Prove that
(i) ( triangle A E F sim triangle C D F )
(ii) ( triangle A B D sim triangle C B E )
(iii) ( triangle A E F sim triangle A D B )
(iv) ( triangle boldsymbol{F} boldsymbol{D} boldsymbol{C} sim triangle boldsymbol{B} boldsymbol{E} boldsymbol{C} )
10
403 57. In a triangle ABC, the sum of the
exterior angles at B and C is equal
to :
(1) 180° + ZBAC
(2) 180° – ZBAC
(3) 180° + 1 ZBAC
(4) 180° + 2 ZBAC
9
404 If the same photograph is printed in different sizes, we say it is
A. Not similar
B. Similar
c. common
D. None
10
405 ( A B C ) is a triangle in which ( angle B=2 angle C . D )
is a point on BC such that AD bisects ( angle B A C ) and ( A B=C D . ) Find ( angle B A C )
A ( .72^{circ} )
( ^{0} )
B. ( 36^{circ} )
( c cdot 108^{circ} )
D. ( 90^{circ} )
9
406 One of the angles of a ( triangle ) is ( 75 . ) If the
difference of the other two other is 35
Find the largest angle of the ( triangle )
9
407 Hence, ( Delta A D C ) and ( Delta B A C ) are similar.
If the above statement is true then
mention answer as 1 , else mention 0 if
false
10
408 n the figure ( Delta A B C, D E | ) ( B C, A(Delta A D E)=48 ) sq.cm., ( frac{A D}{D B}=frac{4}{5} )
Find the area of ( Delta B E C )
A . 60 sq.cm
B. 95 sq.cm
c. 108 sq.cm
D. 135 sa.cm
10
409 ( D ) is a point on the side ( B C ) of a triangle ( A B C ) such that ( angle A D C=angle B A C )
Show that ( C A^{2}=C B . C D )
10
410 n triangle ( A B C, angle B A C=90^{circ}, ) and
( A D ) is its bisector. If ( D E ) is drawn
( perp A C, ) prove that ( D E times(A B+A C)= )
( boldsymbol{A B} times boldsymbol{A C} )
10
411 If in ( triangle A B C ) and ( triangle D E F, frac{A B}{D E}=frac{B C}{F D} )
then they will be similar if
A. ( angle B=angle E )
в. ( angle A=angle D )
c. ( angle B=angle D )
( mathbf{D} cdot angle A=angle F )
10
412 State Pythagoras theorem. 10
413 If two triangles are symmetric, then
they are
A . Equilateral
B. congruent
c. Equal
D. Isosceles
9
414 The ratio of the area of two similar
triangles is ( 9: 16, ) then the ratio of their
corresponding sides will be
10
415 If ( Delta P Q R sim Delta X Y Z, angle Q=50^{circ} ) and
( angle R=70^{circ}, ) then the angle ( angle X+angle Y ) is
equal to:
A ( .70^{circ} )
B. 50
( c cdot 120 )
D. ( 110^{circ} )
10
416 In ( Delta A B C, angle A=30^{circ}, angle B=40^{circ} ) and
( angle C=110^{circ} )
( ln Delta P Q R, angle P=30^{circ}, angle Q=40^{circ} ) and
( angle R=110^{circ} )
Then Is ( Delta A B C cong Delta P Q R ) by ( mathrm{AAA} ? )
A. True
B. False
9
417 59. The ratio of the areas of two isos-
celes triangles having equal ver-
tical angles is 1 : 4. The ratio of
their heights will be
(1) 1 : 2 (2) 3:4
(3) 2:3 (4) 6:7
10
418 In given figure, ( P B ) and ( Q ) a are perpendiculars to segment AB. If PO=5 ( mathrm{cm}, mathrm{Q} mathrm{O}=7 mathrm{cm} ) and Area ( triangle boldsymbol{P O B}= )
( 150 c m^{2} ) find the area of ( triangle Q O A )
10
419 Prove that the line segments joining the midpoints of the sides of a triangle from
four triangles each of which is similar
to the original triangle.
10
420 If ( l, m, n ) are three parallel lines and the
transversal ( t_{1} ) and ( t_{2} ) cut the lines ( l, m, n )
at the points ( A, B, C ) and ( P, O, R ) as
shown in the figure, then
A ( cdot frac{A B}{B C}=frac{P O}{O R} )
B. ( frac{A B}{O R}=frac{B C}{P O} )
c. ( frac{A P}{B O}=frac{B O}{C R} )
D. ( frac{A B}{P O}=frac{A P}{B O} )
10
421 Which of the following statements is true when ( Delta A B C cong Delta D E F )
( mathbf{A} cdot angle A=angle D )
B . ( angle A=angle E )
c. ( angle A=angle F )
D. none of these
9
422 In the given figure, ( 2 A P=5 P B ) and
( 2 C P=5 P D )
Then, ( Delta A C P ) and ( Delta B D P ) are similar.
If the above statement is true then
mention answer as 1 , else mention 0 if
false.
10
423 f ( triangle A B C cong triangle P R Q ) then ( A B=P Q ).If
true enter 1 else 0
9
424 n given figure, IF ( angle A=angle C ), then prove
that ( triangle A O B sim triangle C O D )
10
425 ( triangle A B C sim triangle D E F . ) Explain whether the
two similar triangles may be congruent
as well.
9
426 n right triangle ( A B C, overline{D E} )
( B C, C D=1.5, ) and ( B E=2.0 )
The sine of angle ( theta ) is equal to
( mathbf{A} cdot mathbf{1} )
2
B. 3 4
( c cdot frac{sqrt{2}}{2} )
( frac{sqrt{3}}{2} )
E . 3
( F )
10
427 A right triangle has a hypotenuse of length ( p c m ) and one side of length ( q c m )
( mathbf{f} p-boldsymbol{q}=1, ) find the length of the third
side of the triangle if ( p=5 ) and ( q=4 )
10
428 n given Figure, ( D E | B C . ) If ( A D= )
( boldsymbol{x}, boldsymbol{D} boldsymbol{B}=boldsymbol{x}-boldsymbol{2}, boldsymbol{A} boldsymbol{E}=boldsymbol{x}+boldsymbol{2} ) and ( boldsymbol{E} boldsymbol{C}= )
( x-1, ) find the value of ( x )
10
429 If the area of two similar triangles are equal, then prove that they are
congruent.
10
430 ( E ) and ( F ) are points on the sides ( P Q ) and
( P R ) respectively of a ( Delta P Q R ) For each of
the following cases state whether ( boldsymbol{E F} | boldsymbol{Q} boldsymbol{R} )
(1) ( P E=3.9 c m, E Q=3 mathrm{cm}, P F= )
( mathbf{3 . 6} ) cm and ( boldsymbol{F R}=mathbf{2 . 4} )
(2) ( P E=4 c m, Q E=4.5 mathrm{cm}, P F= )
( 8 mathrm{cm} ) and ( R F=9 mathrm{cm} )
(3) ( P Q=1.28 mathrm{cm} mathrm{PR}= )
( mathbf{2 . 5 6} mathbf{c m}, boldsymbol{P} boldsymbol{E}=mathbf{0 . 1 8} mathbf{c m} ) and ( boldsymbol{P} boldsymbol{F}= )
( mathbf{0 . 3 6} c boldsymbol{m} )
10
431 The same ratio of corresponding sides is referred to as the factor for
polygons.
( A ). scale
B. vernier
c. similar
D. congruence
9
432 Two angles of one triangle are ( 85^{circ} ) and ( 65^{circ} )
is equal to angles of the other. Are they similar? Prove it.
10
433 n given figure ( A B C D ) is a
quadrilateral in which ( P, Q, R ) and ( S ) are
mid-points of the sides. ( A B, B C, C D )
and ( D A . A C ) is a diagonal. Show that:
(i) ( boldsymbol{S R} | boldsymbol{A C} ) and ( boldsymbol{S R}=frac{mathbf{1}}{mathbf{2}} boldsymbol{A C} )
(ii) ( boldsymbol{P Q}=boldsymbol{S} boldsymbol{R} )
(iii) ( P Q R S ) is a parallelogram
10
434 Which of the following statements are true (T) and which are false(F):

The two altitudes corresponding to two equal sides of a triangle need not be equal.
A. True
B. False

10
435 ( triangle P Q R ) is right angled at ( Q, Q X perp )
( boldsymbol{P R}, boldsymbol{X} boldsymbol{Y} perp boldsymbol{R} boldsymbol{Q} ) and ( boldsymbol{X} boldsymbol{Z} perp boldsymbol{P Q} ) are
drawn. Prove that ( boldsymbol{X} boldsymbol{Z}^{2}=boldsymbol{P} boldsymbol{Z} times boldsymbol{Z} boldsymbol{Q} )
10
436 ( mathrm{BE}=mathrm{DF} )
f the above statement is true then
mention answer as 1 , else mention 0 if
false
10
437 In ( triangle A B C, P Q ) is a line segment intersecting ( A B ) at ( P ) and ( A C ) at ( Q ) such that ( mathrm{PQ} | mathrm{BC} ) and ( mathrm{PQ} ) divides ( triangle mathrm{ABC} ) into two parts equal in area. Find ( frac{B P}{A B} ) 10
438 The altitudes of triangle are 12,15 and 20 units. The largest angle in the triangle is:
A ( .75^{circ} )
B. ( 90^{circ} )
( c cdot 120^{circ} )
D. ( 135^{circ} )
10
439 A ( 15 mathrm{m} ) long ladder reached a window 12
m high from the ground on placing it
against a wall at a distance a. Find the distance of the foot of the ladder from
the wall.
10
440 If ( A B=D E, B C=E F ) and ( A C=D F )
then ( Delta A B C_{-}-_{-} Delta D E F )
( A cdot cong )
B. ( approx )
( c cdot t )
D. >
9
441 Which condition can be used to prove
that the given triangles are SAS
congruent?
( A . angle B=angle F, overline{A B} cong overline{D F} ) and ( overline{A C} cong overline{D E} )
B. ( angle A=angle F, overline{A B} cong overline{D C} ) and ( overline{A C} cong overline{D E} )
( mathrm{c} . angle B=angle C, overline{A B} cong overline{D C} ) and ( overline{A C} cong overline{D E} )
D. ( angle B=angle F, overline{A C} cong overline{D C} ) and ( overline{A C} cong overline{D E} )
( E )
10
442 Take two similar shapes. If you slide rotate or flip one of them, does the
similarity remain.
10
443 In a right-angled, triangle the area of the square drawn on the Hypotenuse is ( 289 c m^{2} ) and the area of the
square drawn on the Base is ( 225 mathrm{cm}^{2} )
What will be the area of the square
drawn on the Height of the triangle in
( c m^{2} ? )
10
444 In the figure, ( P Q R S ) is a parallelogram with ( P Q=15 c m ) and ( R Q=10 c m . L ) is
a point on ( R P ) such that ( R L: L P=2: )
3. ( Q L ) produced meets ( R S ) at ( M ) and ( P S )
produced at ( N . ) Find the lengths of ( P N )
and ( boldsymbol{R} boldsymbol{M} )
A. ( P N=15 mathrm{cm} ; R M=10 mathrm{cm} )
B. ( P N=10 mathrm{cm} ; R M=10 mathrm{cm} )
c. ( P N=25 mathrm{cm} ; R M=15 mathrm{cm} )
D. ( P N=25 mathrm{cm} ; R M=10 mathrm{cm} )
10
445 Which minimum measurements do you
require to check if the given figures are
congruent:
Two rhombuses
9

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