# The Triangle And Its Properties Questions

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

#### List of the triangle and its properties Questions

Question NoQuestionsClass
1A tower of height b and a flagstaff on top of tower subtend equal at a point on the ground, distant a from the base of the
tower. The height of the flagstaff is
A ( cdot frac{b^{2}}{a+b} )
B. ( frac{b(a+b)}{a-b} )
c. ( frac{b^{2}}{a-b} )
D. ( bleft(frac{a^{2}+b^{2}}{a^{2}-b^{2}}right) )
7
2Find ( ^{prime} x^{prime} ) for the given figure below7
311. Let PQRS be a rhombus, find x, y.
10.
11.
7
4In ( Delta ) ABC ( angle A=65^{0} ) and ( angle B=25^{circ} C )
Name the hypotenuse.
7
5( angle B C A )7
6Construct a ( triangle A B C ) in which the base
( B C=5 mathrm{cm}, angle B A C=40^{circ} ) and the
median from ( boldsymbol{A} ) to ( boldsymbol{B} boldsymbol{C} ) is ( boldsymbol{6} mathrm{cm} . ) Also
measure the length of the altitude from
( A )
7
7Find the value of the unknown exterior ( x )
in the following diagrams.
7
8In the adjoining figure ( A B= )
( 12 mathrm{cm}, C D=8 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 2
c. ( x ) has only one value and ( x geq 12 )
D. ( x ) cannot be determined with the given information
7
9AD bisects angle ( A ) of triangle ( A B C ) where D lies on BC and angle ( mathrm{C} ) is greater than angle B. then angle ADB is greater than angle ADC.
A . True
B. False
7
10Which of the following sets of side lengths will not form a triangle?
A. ( 11 mathrm{cm}, 10 mathrm{cm}, 11 mathrm{cm} )
в. ( 3 mathrm{m}, 3 mathrm{m}, 3 mathrm{m} )
( mathbf{c} .9 mathrm{mm}, 9 mathrm{mm}, 12 mathrm{mm} )
D. ( 3 mathrm{cm}, 4 mathrm{cm}, 7 mathrm{cm} )
7
11In an isosceles ( triangle A B C ) is the ( A B=A C )
( D ) and ( E ) are points on ( B C ) such that
( B E=C O ) show that ( A D=A E )
7
12( A B C ) is a right angled triangle is which
( angle A=90^{circ} ) and ( A B=A C . ) Find ( angle B ) and
( angle C )
7
13If in two triangles ( A B C ) and ( P Q R ) ( frac{A B}{Q R}=frac{B C}{P R}=frac{C A}{P Q} . ) Write a relation
between the two triangles.
7
14In the given Fig
f ( boldsymbol{A B} | boldsymbol{C D}, boldsymbol{E F} perp boldsymbol{C D} boldsymbol{a n d} angle boldsymbol{G} boldsymbol{E D}= )
( 126^{circ}, ) Find ( angle A G E, angle G E F, angle F G E )
7
15In a ( triangle A B C, ) perpendicular ( A D ) from ( A ) on ( B C ) meets ( B C ) at ( D . ) If ( B D=8 mathrm{cm} )
( D C=2 mathrm{cm} ) and ( A D=4 mathrm{cm}, ) then
A. ( triangle A B C ) is isosceles
B. ( triangle A B C ) is equilateral
c. ( A C=2 A B )
D. ( triangle A B C ) is right angled at ( A )
7
16Find the measure of each angle of an
equilateral triangle.
7
17Find ( y )
( mathbf{A} cdot 45^{circ} )
B ( .60^{circ} )
( c cdot 15^{c} )
D . 20
7
18In an isosceles triangle ( A B C ) with ( A B= )
( A C, B D ) is perpendicular from ( B ) to the side AC. Prove that ( B D^{2}-C D^{2}= )
( 2 C D . A D )
7
19( ln a Delta A B C, I f A C>A B ) and the
bisector of ( angle A ) meets ( B C ) at ( E ), then
A. ( C E>B E )
в. ( C EB E )
D. ( C D<B E )
7
20In the given figure, ( boldsymbol{O} ) is a point in the
interior of ( 1 M ) a square ( A B C D ) such
that ( O A B ) is an equilateral triangle.
Show that ( O C D ) is an isosceles
triangle.
7
21( angle boldsymbol{A}+angle boldsymbol{B}=mathbf{6 5}^{circ}, angle boldsymbol{B}+angle boldsymbol{C}=mathbf{1 4 0}^{circ} )
Find measure of each angle of ( Delta )
7
22If two angles in a triangle are ( 65^{circ} ) and
( 85^{circ}, ) then the third angle is:
( A cdot 30 )
B . 45
( c cdot 60^{circ} )
D. ( 90^{circ} )
7
23In the figure given, lines ( X Y ) and ( M N )
intersect at ( 0 . ) If ( angle P O Y=90^{circ} ) and ( a: b= )
( 2: 3, ) then ( angle X O N ) is equal to
A . ( 126^{circ} )
B . ( 30^{circ} )
( c cdot 90^{circ} )
D. ( 180^{circ} )
7
24Find the name of the triangle.
A. isosceles
B. equilateral
( c . ) scalene
D. acute
7
25( ln Delta mathrm{ABC}, overline{X Y} ) is paralled to ( overline{A C} ) and
divides the triangle into two parts of equal area. Then the ( frac{A X}{A B} ) equals
A ( cdot frac{sqrt{2}+1}{2} )
B. ( frac{2-sqrt{2}}{2} )
c. ( frac{2+sqrt{2}}{2} )
D. ( frac{sqrt{2}-1}{2} )
7
26An obtuse triangle will have one and only one angle.
A . acute
B. obtuse
c. scalene
D. right angled
7
27Find the values of ( x ) and ( y ) in the given
triangles, where ( angle A=70^{circ}, angle B= )
( mathbf{3 6}^{circ}, angle boldsymbol{D}=mathbf{6 8}^{circ} ) and ( angle boldsymbol{F}=mathbf{5 6}^{circ} )
A ( . x=103, y=80 )
B . ( x=105, y=75 )
C ( . x=105, y=87 )
7
28If in a triangle ( boldsymbol{L} boldsymbol{M} boldsymbol{N} )
( angle M=60^{circ}, angle N=60^{circ}, ) find ( angle L )
Mention the kind of triangle also.
7
29In triangle ( A B C, 3 angle A=4 angle B=6 angle C . ) The
smallest angle of the triangle is ( 50^{circ} )
A. True
B. False
7
30A triangle can have:
A. Two right angles
B. Two obtuse angles
C. All angles more than ( 60^{circ} )
D. Two acute angles
7
31( triangle A B C ) is right angled in which ( angle A= )
( 90^{circ} ) and ( A B=A C . ) Find ( angle B ) and ( angle C )
7
32In a triangle, there are 3 different angles.Which of the following statements is/are definitely true about the angles?
(i) Sum of all the angles is ( 100^{circ} )
(ii) One angle is obtuse and other two
are acute and right angle respectively.
(iii) Sum of any two angles is always
less than ( 180^{circ} )
A. (ii) only
B. (ii)and (iii) both
c. (iii) only
D. (i) and (iii) both
7
33It is not possible to construct a triangle
with which of the following sides?
A. ( 8.3 mathrm{cm}, 3.4 mathrm{cm}, 6.1 mathrm{cm} )
B. ( 5.4 mathrm{cm}, 2.3 mathrm{cm}, 3.1 mathrm{cm} )
c. ( 6 mathrm{cm}, 7 mathrm{cm}, 10 mathrm{cm} )
D. ( 3 mathrm{cm}, 5 mathrm{cm}, 5 mathrm{cm} )
7
34Angles opposite to the equal sides of a triangle are equal.ff true enter 1 else 07
35Find the measures of the third angle in
degrees
( 92^{circ}, 27^{circ},—- )
7
36A piece of wire ( 28 mathrm{m} ) long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to maximize the
total area?
7
37Fill in the blank
All ( dots dots dots dots dots ) triangles are similar
A. equilateral
B. scalene
c. isoceles
D. None of these
7
38Verify the points (0,7,-10)(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle.7
39The sum of lengths of any two sides of a triangle is always the
third side.
A. greater than
B. less than
c. equal to
D. none of these
7
40Consider the points ( boldsymbol{P}(mathbf{2},-mathbf{4}) ; boldsymbol{Q}(mathbf{4},-mathbf{2}) boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{R}(mathbf{7}, mathbf{1}) . ) The
points P,Q,R
A. form an equilateral triangle
B. form a right angled triangle
c. form an isosceles triangle which is not equilateral
D. are collinear
7
41n Fig., the side ( Q R ) of ( triangle P Q R ) is produced
to a point ( S ) If the bisectors of ( angle P Q R ) and ( angle P R S )
meet at point ( T, ) then prove that ( angle mathrm{QTR}=frac{1}{2} angle mathrm{QPR} )
7
42( ln triangle A B C, A B=B C=K, A C=sqrt{2} k, ) then
( triangle A B C ) is a :
This question has multiple correct options
A. Isosceles triangle
B. Right-angled triangle
c. Equilateral triangle
D. Right isosceles triangle
7
43If three altitudes of a triangle are equal then the triangle is
A . Right angled
B. Equilateral
c. Isosceles
D. Scalene
7
44( triangle A B C ) is iscoeles in which ( A B=A C )
( operatorname{seg} B D ) and ( operatorname{seg} C E ) are medians show ( operatorname{that} B D=C E )
7
45In the adjoining figure, find the
measure of ( angle B C D )
7
46( ln Delta A B C, A D ) is drawn such that
( Delta A B D ) and ( Delta A C D ) are equal in area
then, the AD is
A. any segment drawn from A to BC
B. the bisector of ( angle B A G )
c. A median of ( Delta A B C )
D. None of these
7
47Find the values of ( x ) and ( y ) in the following figures
A ( . x=80^{0} ; y=120 )
В . ( x=50^{0} ; y=30^{0} )
C . ( x=50^{circ} ; y=130^{circ} )
D . ( x=20^{0} ; y=10^{0} )
7
48( A ) and ( B ) are two fixed points in a plane. If Pis a moving point in the plane such that ( P A=P B, ) then the:
A. locus of P is the line AB itself.
B. locus of P is a line parallel to AB
c. point P always makes equilateral triangles with A, B.
D. triangle PAB is isosceles for all positions of P.
7
49The point 0 lies inside a triangle ( P Q R ) such that ( Delta O P Q, Delta O Q R ) and ( Delta O R P ) are equal in area. Then,the point 0 is
called as
A. incentre
B. centroid
c. circumcentre
D. orthnocentric
7
50( boldsymbol{P Q}=boldsymbol{Q} boldsymbol{R}=boldsymbol{P S} . ) Calculate the size of
the labelled angles.
B . (b) ( a=42^{circ}, b=48^{circ}, c=69^{circ}, d=111^{circ} )
C . (c) ( a=45^{circ}, b=45^{circ}, c=67.5^{circ}, d=112.5^{circ} )
D cdot (d) ( a=50^{circ}, b=40^{circ}, c=65^{circ}, d=115^{circ} )
7
51ldentify the largest side of the triangle. f the angles are given as:
( angle A=50^{circ}, angle B=10^{circ} ) and ( angle C=22^{circ} )
( A cdot overline{A B} )
8. ( overline{B C} )
( c cdot overline{C A} )
( D . overline{A C} )
7
52( ln Delta A B C, 2 angle A=3 angle B=6 angle C . ) Find
( frac{angle A+angle B}{angle C} times angle B )
7
53In the figure, ( angle P Q R=angle P R Q, ) then
prove that ( angle P Q S=angle P R T )
7
54In the figure given below, the value of
( (angle x+angle y) ) is :
4. ( 110^{circ} )
B . ( 100^{circ} )
( c cdot 120 )
D. ( 60^{circ} )
7
55The sides of a triangle with positive area have lengths 5,7 and ( a . ) The sides of a second triangle with positive area
have lengths 5,7 and ( b ) Which of the following is NOT a possible
value of ( |boldsymbol{a}-boldsymbol{b}| ? )
A . 3
B. 5
( c cdot 7 )
D. 10
7
56Angles opposite to equal sides of an isosceles triangle are equal. Prove this
result
7
57Assertion: An isosceles ( Delta ) is right angled.
Reason ( : angle A=angle B=45^{circ} ) and ( angle C= )
( mathbf{9 0}^{circ} )
Which of the following statement is
true?
A. A is true and ( mathrm{R} ) is the correct explanation of
B. A is true and R is not the correct explanation of
c. A is false
D. None of these
7
58In a triangle, the Sum of two angles is 118 and their difference is ( 32 . ) find each
angle of the triangle.
A. 75,43 and 62
B. 85,33 and 62
c. 65,53 and 62
D. none of the above
7
59In the triangle, find the measures of the
angles.
A ( cdot 40^{circ}, 60^{circ}, 80^{circ} )
B ( cdot 45^{circ}, 60^{circ}, 75^{circ} )
c. ( 45^{circ}, 45^{circ}, 90^{circ} )
D. ( 50^{circ}, 60^{circ}, 70^{circ} )
7
60In the figure at right, ( A B ) and ( C D ) are
straight lines through the centre 0 of a
circle. If ( angle A O C=98^{circ} ) and ( angle C D E= )
( mathbf{3 5}^{circ} )
Find (i) ( angle D C E )
(ii) ( angle A B C )
7
61If ( left(x_{1}, y_{1}, z_{1}right) ) and ( left(x_{2}, y_{2}, z_{2}right) ) are two vertices and ( (alpha, beta gamma) ) is the centroid of a triangle, find the third vertex of the triangle.7
62If ( Delta A B C ) is an equilateral triangle of side
( a ) and ( mathrm{D} ) is a point on ( B C ) such that ( B D=frac{1}{3} B C ) then the prove that ( A D= )
( frac{sqrt{7} a}{3} )
7
63Using triangle inequality theorem check whether the given side lengths
( boldsymbol{a}=mathbf{3}, boldsymbol{b}=mathbf{5} ) and ( boldsymbol{c}=mathbf{1} ) will form a
triangle or not
7
64n Fig. ( angle boldsymbol{P}=mathbf{5} mathbf{2}^{circ} boldsymbol{a} boldsymbol{n} boldsymbol{d} angle boldsymbol{P} boldsymbol{Q} boldsymbol{R}=boldsymbol{6} boldsymbol{4}^{circ}, ) i
( Q O ) and ( R O ) are the angle bisectors of
( angle P Q R ) and ( angle P R Q ) respectively, then
find the values of ( angle x ) and ( angle y )
7
65Find the values of ( x ) and ( y ) in the figure
given below
7
66If the two angles of a triangle are unequal, then the smaller angle has the
side opposite to it.
A. Smaller
B. Larger
C. May be smaller may be larger
D. The side will be equal to one of the opposite sides
7
67If each side of an equilateral triangle is
8, calculate the length of the altitude.
A . 1.73
B . 2
c. 3.46
D. 4
E . 6.93
7
68The centroid of the triangle whose vertices are (4,-8),(-9,7) and (8,13) is
в. (1,3)
c. (1,5)
(年. (1,5)
D. (1,9)
7
69In the given figure shown ( boldsymbol{P Q R S} ) is a
square and ( S R T ) is an equilateral
triangle then state whether ( angle Q T R= )
( 15^{circ} ) is true/false.
A. True
B. False
7
70( Delta A B C ) is an isosceles right triangle
with area ( P . ) The radius of the circle that
passes through the point ( A, B ) and ( C ) is:
( A cdot sqrt{P} )
B. ( sqrt{frac{P}{2}} )
c. ( frac{sqrt{P}}{2} )
D. ( sqrt{2 P} )
7
71In the following figure, ( S ) is any point on
side ( mathrm{BC} ) of ( Delta mathrm{ABC} Delta mathrm{ABC} ). Then ( mathrm{AB}+mathrm{BC}+mathrm{CA} )
२२४S
A. True
B. False
7
72( Delta A B C ) is an isosceles triangle with
( A B=A C . A D ) bisects ( angle A . ) Prove that
( angle B=angle C )
7
73The perimeters of an equilateral
triangle and a square are same The area of triangle : area of square is
A . 1: 1
в. ( 4: 3 sqrt{3} )
c. 4: 3
D. 3: 2
7
74In a triangle ( A B C, angle A+angle B=144^{circ} ) and ( angle A )
( +angle C=124^{circ} )
Calculate each angle of the triangle.
A ( cdot A=88^{circ}, B=56^{circ} ) and ( C=36 )
B. A ( =78^{circ}, B=66^{circ} ) and ( C=36^{circ} )
C. ( A=58^{circ}, B=86^{circ} ) and ( C=36^{circ} )
D. ( A=88^{circ}, B=26^{circ} ) and ( C=66^{circ} )
7
75Identify the largest angle of the triangle.
( A )
в.
( c )
( D )
7
76The number of lines of symmetry in a scalene triangle is
A. 0
B.
( c cdot 2 )
( D )
7
77n the given figure, ( triangle A B C ) is an equilateral triangle and ( square A W X B ) and
( square A Y Z C ) are two squares. The value of ( frac{1}{10}(angle Z X A) ) is:
7
78In the adjoining figure find the values of
( x ) and ( y )
7
79The combined equation of the three sides of a triangle is
( left(x^{2}-y^{2}right)(2 x+3 y-6) . ) If the point
( (0, a) ) lies in the interior of this triangle
then
A ( .-2<alpha<0 )
B. ( -2<alpha<2 )
c. ( 0<alpha<2 )
D. ( alpha geq 2 )
7
80An equilateral triangle has lines of symmetry.
A. 0
B.
( c cdot 3 )
( D )
7
81In the given figure, find
( angle boldsymbol{P} boldsymbol{N} boldsymbol{M} )
7
82Construct ( triangle A B C ) with ( A B= )
( 5 c m, B C=5 c m ) and ( A C=5 c m )
7
83n ( triangle boldsymbol{P Q R}, boldsymbol{P Q}=boldsymbol{P R}, angle boldsymbol{Q}=mathbf{6 5}^{circ} ) then
find ( angle P )
7
84( A D, B E ) and ( C F, ) the altitude of
( Delta A B C ) are equal. Then
( mathbf{A} cdot A C=B C )
B . ( A D=A B )
c. ( A B=C F )
D. None of these
7
85Identify the triangle as equiangular, acute, obtuse or right.
Answer: Obtuse
Mark answer as 1 if true else 0 if false
7
86n given figure the interior opposite
angles of the exterior angle ( angle A C D ) are:
( A cdot angle B, angle C )
в. ( angle A, angle C )
( mathbf{c} cdot angle A, angle B )
D. ( angle B, angle E )
7
87( P ) is any point inside the triangle ( A B C ) Prove that: ( angle B P C>angle B A C )7
88A triangle has side lengths of 6 inches
and 9 inches. If the third side is an integer, calculate the minimum possible perimeter of the triangle (in inches).
A . 4
B. 15
c. 8
D. 19
E . 29
7
89The sides of a triangle are ( mathbf{5 0} mathrm{cm}, mathbf{7 8} mathrm{cm} ) and ( 112 mathrm{cm} . ) The smallest
altitude is…
( mathbf{A} cdot 20 mathrm{cm} )
B. ( 30 mathrm{cm} )
c. ( 40 mathrm{cm} )
D. ( 50 mathrm{cm} )
E. None of these
7
90Name the type of following triangle.
( Delta X Y Z ) with ( m angle Y=90^{circ} ) and ( X Y=Y Z )
7
91In an isosceles triangle, the base angles
are equal. The vertex angle is ( 40^{circ} . ) What
are the base angles of the triangle? (Remember, the sum of three angles of
a triangle is ( 180^{circ} ) ).
7
92Compute the value of ( x ) in the figure
given
7
93( mathbf{n} Delta boldsymbol{P} boldsymbol{Q} boldsymbol{R}, angle boldsymbol{P}=mathbf{2} boldsymbol{x}+mathbf{1}^{circ}, angle boldsymbol{Q}=mathbf{3} boldsymbol{x}+ )
( mathbf{3}^{circ} boldsymbol{a} boldsymbol{n} boldsymbol{d} angle boldsymbol{R}=boldsymbol{x}+boldsymbol{2}^{circ} . ) What is the value
of x?
7
94Calculate ( angle B A C )
( A cdot 90^{circ} )
B ( .50^{circ} )
( c cdot 60^{circ} )
( mathbf{D} cdot 80 )
7
95Find the value of the unknown interior
angle ( x ) in the following figure.
7
96The angles in a Quadrilateral are ( x, 5 x, 2 x+10, x+80 ) Find ( x )
A . 30
B . 18
( c .35 )
D. 42
7
97Prove that the lines represented by ( 3 x^{2}-8 x y-3 y^{2}=0 ) and ( x+2 y=3 )
From the sides of an isosceles right angled triangle.
7
98In the figure above, points ( A, D, B, ) and ( G )
are collinear. If ( angle C A D ) measures ( 76^{circ} )
( angle B C D ) measures ( 47^{circ}, ) and ( angle C B G )
measures ( 140^{circ}, ) find the degree
measure of ( angle boldsymbol{A} boldsymbol{C} boldsymbol{D} )
A ( cdot 12^{circ} )
B . ( 14^{circ} )
( c cdot 17 )
( mathbf{D} cdot 36 )
( E cdot 43 )
7
99In the adjacent figure value of ( x )
( mathbf{A} cdot 67^{circ} )
В. 157
( mathrm{c} cdot 179^{circ} )
( mathbf{D} cdot 360^{circ} )
7
100What is the largest side of the triangle?
A. ( overline{A B} )
в. ( overline{B C} )
( c cdot overline{C A} )
D. ( overline{A C} )
7
101Two triangles having the same base (or
equal bases) and equal areas lie
between the same parallels
7
102In the adjacent triangle ( A B C ), find the
value of ( x ) and calculate the measure of
all the angles of the triangle.
7
103Find ( angle A ) in ( triangle A B C ) in which ( angle B= )
( mathbf{6 0}, angle boldsymbol{C}=mathbf{4 5}^{mathbf{0}} )
7
104If a triangle ( P Q R ) has been constructed ( operatorname{taking} Q R=6 mathrm{cm}, P Q=3 mathrm{cm} ) and
( P R=4 mathrm{cm}, ) then the correct order of
the angle of triangle is
A. ( angle P<angle Qangle Qangle Q>angle R )
D. ( angle Pangle R )
7
105Which of the following sets of side lengths form a triangle?
A. ( 4 mathrm{m}, 3 mathrm{m}, 11 mathrm{m} )
B. 7 ( mathrm{mm}, 4 mathrm{mm}, 4 mathrm{mm} )
( c .3 mathrm{cm}, 1.23 mathrm{cm}, 5 mathrm{cm} )
D. 3 ( m, 10 mathrm{m}, 8 mathrm{m} )
7
106The sides of a triangle ( A B C ) are
positive integers. The smallest side has
length ( l . ) What of the following
statements is true?
A. The area of ABC is always a rational number
B. The area of ABC is always an irrational number
c. The perimeter of ABC is an even integer.
D. The information provided is not sufficient to conclude any of the statements A, B or C above
7
10758. If I be the incentre of A ABC
and ZB = 70° and ZC = 50°.
then the magnitude of ZBIC is
(1) 130°
(2) 60°
(3) 120° (4) 105°
7
108The angles of triangle are ( x, 5 x, 9 x ) then
find the angles of triangle.
7
109In an equilateral triangal ( A B C, D ) is a point on side ( mathrm{BC} ) such that ( B D=frac{1}{3} B C )
Prove that ( 9 A D^{2}=7 A B^{2} )
7
110In ( triangle A B C, ) if ( m angle B=90 & A C=10 )
then length of median ( boldsymbol{B} boldsymbol{M}= )
( A cdot 6 )
B. ( 5 sqrt{2} )
c.
D. 8
7
111Find the measures of the missing angle
in the given triangle.
A ( cdot 30^{circ} )
В. ( 60^{circ} )
( c cdot 90^{circ} )
D. ( 12^{circ} )
7
112Based on the sides, classify the
following triangles
7
113Form the given figure, find the values of
( x ) and ( y ) respectively.
A ( cdot 47^{circ}, 66^{circ} )
B ( cdot 66^{circ}, 48^{circ} )
( mathbf{c} cdot 68^{circ}, 47^{circ} )
D. ( 47^{circ}, 68^{text {? }} )
7
114( Delta A B C ) is an isosceles right triangle
with area ( P ).The radius of the circle that
passes through the point ( A, B ) and ( C ) is
( A cdot sqrt{P} )
B. ( sqrt{frac{P}{2}} )
c. ( frac{sqrt{P}}{2} )
D. ( sqrt{2 P} )
7
115In each of the following state if the statement is true (T) or false (F):
Every acute triangle is equilateral.
A. True
B. False
7
116The largest angle of a triangle is twice
the sum of the other two and the
smallest one is one fourth of the largest,
the angles are
A ( cdot 120^{0}, 40^{0}, 20^{0} )
В . ( 120^{0}, 30^{0}, 30^{0} )
c. ( 90^{circ}, 45^{circ}, 45^{circ} )
D. ( 90^{0}, 60^{0}, 30^{0} )
7
117The measure of the third angle of the
triangle. ( in degrees )
( 110^{circ}, 23^{circ},— )
7
118The angles of a triangle are in the ratio
( 2: 3: 4 . ) Find the angles.
7
119Show that the following points form an equilateral triangle. ( (sqrt{3}, 2),(0,1) ) and (0,3)7
120The sides of a triangle (in cm) are given
below:

In which case, the construction of ( triangle ) is
not possible?
( A cdot 8,7,3 )
B. 8,6,4
c. 8,4,4
D. 7,6,5

7
121Find the length of the longest side of the triangle formed by the line ( 3 x+4 y= )
12 with the coordinate axes.
7
122In the adjacent figure, it is given that
( A B=A C, angle B A C=36^{circ}, angle A D B=45^{circ} )
and ( angle A E C=40^{circ} . ) Find ( angle A C B )
7
123( A B C ) is an equilateral triangle of side 6 cm. If a circle of radius ( 1 mathrm{cm} ) is moving inside and along the sides of the triangle, then locus of the centre of the circle is an equilateral triangle of side
A. ( 5 mathrm{cm} )
B. ( 4 mathrm{cm} )
c. ( (6-2 sqrt{3}) ) ст
D. ( (3+sqrt{3}) ) cm
7
124As shown in the figure ( boldsymbol{A C}= )
( boldsymbol{C D}, angle boldsymbol{C A B}-angle boldsymbol{A B C}=mathbf{3 0}^{circ} . ) Then
( angle B A D ) has a measure of:
A ( cdot 15^{circ} )
B. 30
( c cdot 20 )
( left(22 frac{1}{2}right)^{circ} )
7
125In ( Delta A B C, A B=A C . D, E ) and ( F ) are
mid-points of the sides ( B C, C A ) and
( A B ) respectively. then, ( : A D ) is
perpendicular to ( boldsymbol{F C} ) If the above statement is true then
mention answer as 1 , else mention 0 if false
7
126Show that the following points form an isosceles triangle. (2,3),(5,7) and (1,4)7
127If 6,10,14 are the sides of a triangle, then its obtuse angle is
A ( cdot 110^{circ} )
B . ( 120^{circ} )
( mathrm{c} cdot 135^{circ} )
D. ( 115^{circ} )
7
128( ln Delta mathrm{ABC} angle A=40^{circ} ) and ( angle B=60^{circ} . ) The
longest side of triangle ( A B C ) is ( A B )
A. True
B. False
7
129Assertion
If two triangles have same perimeter, then they are congruent.
Reason
If under a given correspondence, the three sides of one triangle are equal to the three sides of the other triangle,
then the two triangles are congruent.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason are correct
7
130The ( _{–}-_{-}-_{-}- ) of a triangle is the perpendicular from a vertex to the opposite side.
A. centroid
B. altitude
c. midpoint
D. orthocenter
7
131n the ( triangle A B C ),we have ( angle A>angle B>angle C )
then determine the shortest and the
longest side of the triangle.
A. Shortest side is ( A B ) and the longest side is ( B C )
B. Shortest side is ( B C ) and the longest side is ( A B )
C. Shortest side is ( A B ) and the longest side is ( A C )
D. Shortest side is ( A C ) and the longest side is ( B C )
7
132In a ( triangle A B C, ) measure ( angle A=35^{0} )
measure ( angle B=65^{0}, ) find measure ( angle C )
A ( cdot 60^{circ} )
В. ( 70^{circ} )
( c .80^{circ} )
D. None of these
7
133ABCD is a rectangle. If ( A B P ) and ( B C Q ) are
equilateral triangle, ( angle P B Q=dots )
A ( cdot 65 )
B. 75
( c cdot 60 )
D. ( 90^{circ} )
7
134If ( A D, B E ) and ( C F ) are the medians of ( angle )
ABC, then which one of the following
statements is
correct?
( mathbf{A} cdot(A D+B E+C F)=(A B+B C+C D) )
B ( cdot(A D+B E+C D)>frac{3}{4}(A B+B C+C A) )
( mathbf{c} cdot(A D+B E+C F)>frac{3}{4}(A B+B C+C A) )
D. ( (A D+B E+C F)=frac{1}{2}(A B+B C+C A) )
7
135( Delta A B C ) fig. ( 2, angle x+angle y+angle z ) is equals
A ( cdot 120^{circ} )
B. ( 180^{text {? }} )
( c cdot 240 )
D. 360
7
136The sides of a triangle are equal and have equations ( 2 x-y=0,3 x+y=0, x )
( 3 y+10=0, ) respectively find the equation of three medians of the triangle and verify that they are concurrent
7
137Mark the correct alternative of the
following.
n figure, if ( A B | C D, ) then the values of ( x )
and y are?
A. ( x=106, y=307 )
B. ( x=307, y=106 )
c. ( x=107, y=306 )
D. ( x=105, y=308 )
7
138Find the value of ( boldsymbol{X} )
( mathbf{A} cdot 60^{circ} )
B. ( 70^{circ} )
( mathbf{c} cdot 80^{circ} )
( mathbf{D} cdot 90^{circ} )
7
139In an equilateral triangle of side ( 3 sqrt{3} ) find the length of the altitude.7
140The area of an isosceles triangle, each of whose equal sides is ( 13 mathrm{cm} ) and
whose base is ( 24 c m ) is
( mathbf{A} cdot 60 mathrm{cm}^{2} )
B. ( 55 mathrm{cm}^{2} )
c. ( 50 mathrm{cm}^{2} )
D. ( 40 mathrm{cm}^{2} )
7
141n figure 1, ( angle P Q R ) is :
( mathbf{A} cdot 40^{circ} )
B ( .50^{circ} )
( c cdot 30 )
D. 105
7
142A circular park of radius ( 20 m ) is
situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.
7
143Origin is the centre of circle passing through the vertices of an equilateral triangle whose median is of length ( 3 a )
then equation of the circle is?
A ( cdot x^{2}+y^{2}=a^{2} )
B . ( x^{2}+y^{2}=2 a^{2} )
c. ( x^{2}+y^{2}=3 a^{2} )
D. ( x^{2}+y^{2}+4 a^{2} )
7
144In ( triangle A B C, angle A=45^{circ} ) and ( angle B=65^{circ} )
Name the side of the triangle which is
shortest
7
145f 0 is a point within a quadrilateral
( A B C D, ) prove that ( ; O A+O B+O C+ )
( O D>A C+B D )
7
146Find the length of altitude through ( A ) of
the triangle ( A B C, ) where ( A equiv )
( (-mathbf{3}, mathbf{0}), boldsymbol{B} equiv(mathbf{4},-mathbf{1}), boldsymbol{C} equiv(mathbf{5}, mathbf{2}) )
7
147ABC is an equilaterial triangle with side
( a . A ) point ( P ) is taken inside ABC. The sum of lengths of perpendiculars from to all the sides is
A. greater than altitude from A to BC
B. less than altitude form A to BC
c. equal to altitude from A to BC
D. can’t he determined
7
148n ( triangle A B C ) what is sum of the angles a ( + )
( b+c+d+e+f+g+h+i ? )
( 4 cdot 360 )
3.540
( c cdot 600 )
cannot be determine
7
149triangle has three different sides.
A. Equilateral
B. Isosceles
c. Right angled
D. Scalene
7
150If ( boldsymbol{A}(mathbf{0}, mathbf{3}), boldsymbol{B}(mathbf{0}, mathbf{0}), boldsymbol{C}(mathbf{4} . mathbf{0}) ) be the
vertices of a triangle. Then the triangle is
A. isosceles
B. equilateral
c. right angled
D. none of these
7
151Construct an equilateral triangle, given its side ( =3 ) and justify the
construction.
7
152In a triangle, the sum of two angles is
( 118^{circ} ) and their difference is ( 32^{circ} ; ) find
each angle of the triangle.
A ( cdot 75^{circ}, 43^{circ} ) and ( 62^{circ} )
B . ( 70^{circ}, 48^{8} ) and ( 62^{circ} )
c. ( 85^{circ}, 43^{circ} ) and ( 52^{circ} )
D. ( 75^{circ}, 53^{circ} ) and ( 52^{circ} )
7
153If the hypotenuse of a right angled triangle is ( 41 mathrm{cm} ) and the area of the
triangle is 180 sq ( c m ), then the
difference between the lengths of the triangle must be
( mathbf{A} cdot 22 mathrm{cm} )
B. ( 25 mathrm{cm} )
( c cdot 27 mathrm{cm} )
D. ( 31 mathrm{cm} )
7
154Prove that each angle of an equilateral
triangle is ( 60^{circ} )
7
155Let ( l ) be the length of each equal side of
an isosceles triangle. If the length of each equal side is doubled, keeping its height unchanged, then the difference of the squares of bases of the new triangle and the given triangle is
( mathbf{A} cdot mathbf{0} )
B. ( 4 l^{2} )
( mathrm{c} cdot 9 l^{2} )
D. ( 12 l^{2} )
7
156In any triangle, the side opposite to the larger (greater) angle is longer
A. True
B. False
7
157The measures of angles of a triangle are in the ratio ( 1: 2: 3 . ) Determine the
measures of smallest angle of the
triangle.
7
158Show that the points ( A(1,2), B(1,6), C(1+ ) ( 2 sqrt{3}, 4) ) are vertices of an equilateral triangle.7
159In an isosceles triangle, the vertex angle
is ( 50^{circ} . ) What are the base angles of the
triangle?
7
160In a triangle ( A B C, angle A-angle B=15^{circ} ) and
( angle B-angle C=30^{circ}, ) find ( angle A, angle B ) and ( angle C )
7
161O is any point in the interior of ( Delta A B C ). Prove that
( A B+B C+C A>O A+O B+O C )
7
162PQ is the diameter of given circle and
( angle P R O=35^{circ} . ) Then ( angle R O Q ) equals
being the center):
A ( cdot 40^{circ} )
B. ( 35^{circ} )
( c cdot 105^{circ} )
D. 70
7
163If the bisectors of the angles ( angle A B C ) and
( angle A C B ) of a triangle ( A B C ) meet at a point
0, then Prove that ( angle B O C=90^{circ}+ )
( frac{1}{2} angle B A C )
7
164If all the angles of a triangle measure
less than ( 90^{circ} ), then such a triangle is
called
A. Right angled triangle
B. Obtuse angled triangle
c. Acute angled triangle
D. None of these
7
165In an equilateral triangle of side ( 24 mathrm{cm} )
a circle is inscribed touching its sides. Find the area of the remaining portion
of the triangle.
7
166Number of interior angles formed in the
triangle are
A . 1
B. 2
( c .3 )
( D )
7
167If one angle of a ( Delta ) is equal to the sum of
the other two, the triangle is
A. isosceles
B. equilateral
c. right angled
D. ordinary
7
168In a triangle ( A B C, angle A-angle B=30^{circ} ) and ( angle A )
( angle C=42^{circ} ; ) find angle ( A )
A ( cdot A=84^{circ} )
B ( cdot A=64 )
C ( cdot A=95 )
D. ( A=92^{circ} )
7
169An isosceles triangle contains three
angles that measure ( 40^{circ}, x^{circ}, ) and ( y^{circ} )
Which of the following CANNOT be true?
A ( . x=y )
B. ( x=50 )
c. ( x-y=60 )
D. ( x=70 )
E . ( x=100 )
7
170How many angles in the given figure are less than a right angle?
( k )
A .2
B. 3
( c cdot 4 )
D. 5
7
171The angles of a triangle are in the ratio
( 3: 4: 5 . ) Find the smallest angle.
7
172In figure if ( D E | B C ), then find the ratio
of ( a r .(triangle A D E) ) and ( a r .(triangle D E C B) . ) Also ( D E=6 mathrm{cm}, B C=12 mathrm{cm} )
7
173Which of the following options is
INCORRECT?
( mathbf{A} cdot angle 1=angle 3 )
B . ( angle 1+angle 4+angle 5=180^{circ} )
( mathbf{c} cdot angle 8=angle 6 )
D. ( angle 1+angle 3=180^{circ} )
7
174If ( C E ) is parallel to ( B D ) in the given
figure, then the value of ( x ) will be
A . 45
В. 75
( c .30 )
D. 85
7
175In a ( Delta A B C, angle A B C=angle A C B ) and the
bisectors of ( angle A B C ) and ( angle A C B )
intersect at 0 such that ( angle B O C=120^{circ} )
Show that ( angle A=angle B=angle C=60^{circ} )
7
176In ( triangle A B C, ) bisector of ( angle A ) and ( angle B )
intersect at point ( O . ) If ( angle C=70^{circ} ) what
is the value of ( angle A O B ? )
7
177Find the unknown angles marked in the
following figure.
7
178The perimeter of a triangle ( A B C ) is ( 27 mathrm{cm} ) and the ratio between the
lengths of its sides is 2: 3: 4 find the
sides.
A .6,9,12
в. 3,6,9
c. 3,4,5
D. None
7
179Show that no triangle has two sides each shorter than its corresponding altitude (from the opposite vertex).7
180Angles of a triangle are in the ratio 4: 6
5. The triangle is :
A. an acute angles triangle
B. an obtuse angled triangle
c. right angled triangle
D. isosceles triangle
7
181Take any point ( O ) in the interior of a
( triangle P Q R ) Is ( O R+O P>R P ? )
7
182The side of an equilateral triangle is ( 20 sqrt{3} mathrm{cm} . ) The numerical value of the
radius of the circle circumscribing the triangle is:
( mathbf{A} cdot 20 mathrm{cm} )
в. ( 20 sqrt{3} mathrm{cm} )
c. ( 20 pi c m )
D. ( frac{20}{pi} )
7
183From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. The lengths of the perpendiculars are ( 14 mathrm{cm}, 10 mathrm{cm} ) and ( 6 mathrm{cm} )

The area of the triangle is ( 300 sqrt{m} mathrm{cm}^{2} )
where ( m ) is

7
184In the given figure ( boldsymbol{alpha} ) and ( boldsymbol{beta} ) are
measured in degrees. Which one of the
following statements is not correct?
( mathbf{A} cdot beta>alpha )
( mathbf{B} cdot sec beta=2 )
( mathbf{C} cdot tan 3 alpha=sqrt{3} )
D ( cdot sin (beta-alpha)=frac{1}{sqrt{2}} )
7
185In the given figure find ( O C )7
18667. Let O be the in-centre of a trian-
gle ABC and Dbe a point on the
side BC of AABC, such that OD
I BC. If ZBOD= 15°, then ZABC
(1) 75°
(3) 150°
(2) 45°
(4) 90°
7
187Two angles of a triangle measure ( 52^{circ} )
and ( 72^{circ} . ) Find the measure of the third
angle of the triangle.
A ( .36^{circ} )
B . 46
( c cdot 56^{circ} )
D. ( 66^{circ} )
7
188In each case; given below, find the value of ( x ) in terms of ( a, b ) and ( c )
A ( . x=180^{circ}-a-b+c )
B . ( x=180^{circ}+b+a+c )
c. ( x=180^{circ}+b-a-c )
D. ( x=180^{circ}+a+b-c )
7
189A triangle cannot have more than right angle(s).
A. one
B. two
c. three
D. zero
7
190In the figure the sides ( B C, C A ) and ( A B )
of a ( triangle A B C ) have been produced to ( D, E )
and ( F ) respectively. If ( angle A C D=105^{circ} )
and ( angle E A F=45^{circ}, ) find all the angles of
the ( triangle A B C )
7
191AD is the bisector of ( angle A ) of
( Delta A B C, A C=4.2 c m, D C= )
( 6 c m, B C=10 c m, ) then ( A B ) is:
( mathbf{A} cdot 2.8 mathrm{cm} )
B. 3 cm
( mathrm{c} .3 .5 mathrm{cm} )
D. none of these
7
1925. In Fig, determine ZP+ 2Q+ZR+
S+ZT.
54
p
10
7
193For ( Delta A B C, ) find the measure of ( angle A C D )7
194( angle C E D )7
195If the triangle ( A B C ) in the question 7 above is revolved 7 above is resolved the side ( 5 mathrm{cm}, ) then find the volume of the solid so obtained Find also ratio of the
volumes of the two solids obtained in
Questions 7 and 8
7
196Find the length of the altitude of the
hypotenuse.
( A cdot 5.12 )
3. 5.25
.5 .34
4
5.6
7
197If the equal sides of an isosceles
triangle ( A B C ) are produced, prove that the exterior angles so formed are obtuse,say ext ( angle B, ) and find ( e x t angle B- )
( angle A )
7
198Find the values of ( angle a angle b ) in the figure
given below.
7
199If the bisector of the exterior vertical
angle of a triangle is parallel to the base, show that the triangle is isosceles.
7
200If ( A, B ) and ( C ) are interior angles of a
triangle ( A B C, ) then show that tan ( left(frac{boldsymbol{A}+boldsymbol{B}}{mathbf{2}}right)=cot frac{boldsymbol{C}}{2} )
7
201How many triangles can be drawn having its angles as ( 53^{circ}, 64^{circ} ) and ( 63^{circ} ? )
( mathbf{A} cdot mathbf{1} )
B. 2
c. None
D. More than 2
7
202( A B C ) is an isosceles triangle in which ( A B )
( =A C A ) circle through ( B ) touches side ( A C )
at its mid point ( mathrm{D} ) and intersects ( mathrm{AB} ) at
( P, ) then ( A P ) will be-
( A cdot 2 A D )
B ( cdot frac{1}{2} A B )
c. ( frac{1}{4} A B )
D. ( frac{1}{2} A C )
7
203If two vertices of an isosceles triangle ( operatorname{are}(2,0) ) and (2,5) and length of the equal sides is 3 then the third vertex is
A ( .(2,6) &(-5,3) )
B. (8,3)( &(5,1) )
( ^{mathbf{C}} cdotleft(2 pm frac{sqrt{11}}{2}, frac{5}{2}right) )
D. ( left(3 pm frac{sqrt{14}}{2}, frac{7}{2}right) )
7
204Find the vertex of the median ( boldsymbol{M}_{mathbf{1}} )
( mathbf{A} cdot A )
B. ( B )
( c cdot C )
( D . D )
7
205In figure ( -1, ) which of the following
statement is true?
( mathbf{A} cdot angle B=angle C )
B. ( angle B ) is the greatest angle in triangle
( mathrm{c} . angle B ) is the smallest angle in triangle
D. ( angle A ) is the smallest angle in triangle
7
206If length of the largest side of a triangle is ( 12 mathrm{cm} ) then other two sides of triangle
can be :
( A cdot 4.8 mathrm{cm}, 8.2 mathrm{cm} )
B. 3.2 ( mathrm{cm}, 7.8 mathrm{cm} )
c. ( 6.4 mathrm{cm}, 2.8 mathrm{cm} )
D. ( 7.6 mathrm{cm}, 3.4 mathrm{cm} )
7
207The angles of a triangle are in the ratio
of ( 1: 2: 3 . ) Find the measure of each
angle of the triangle
A ( cdot 30^{circ}, 60^{circ}, 90^{circ} )
B . ( 20^{circ}, 60^{circ}, 90^{circ} )
c. ( 30^{circ}, 30^{circ}, 90^{circ} )
D. None of these
7
208Prove that the median from the vertex of
an isosceles triangle is the bisector of vertical angle.
7
209Two sides of a triangle are of lengths 5 ( mathrm{cm} ) and ( 1.5 mathrm{cm}, ) then the length of the third side of the triangle cannot be
( mathbf{A} cdot 3.6 mathrm{cm} )
B. ( 4.1 mathrm{cm} )
c. ( 3.8 mathrm{cm} )
D. ( 3.4 mathrm{cm} )
7
210State the property that is used in each of the following statements?
If ( angle mathbf{4}=angle mathbf{5}=mathbf{1 8 0}^{circ} ) then ( boldsymbol{a} | boldsymbol{b} )
7
2110 is a point that lies in the interior of ( Delta A B C . ) Then ( 2(O A-O B-O C)> )
Perimeter of ( Delta A B C )
A. True
B. False
7
212( ln Delta A B C, ) if ( A B>B C ) then :
( mathbf{A} cdot angle Cangle A )
( mathbf{D} cdot angle A=angle B )
7
213The height of an equilateral triangle of side ‘a’ is given by
A ( cdot frac{a sqrt{2}}{2} )
B. ( frac{a sqrt{3}}{2} )
c. ( frac{a sqrt{3}}{4} )
D. ( frac{a sqrt{2}}{3} )
7
214( A B C ) is an isosceles triangle such that
( A B=A C . D ) is the mid point of ( A C . A )
circle is drawn taking ( B D ) as diameter which intersects ( A B ) at point ( E ).then
( boldsymbol{A C}=mathbf{3 A E} )
A. True
B. False
7
215In a right-angled triangle ( A B C ) with
( angle C=90^{circ} ) and ( angle A=2 angle B, angle B ) is
A ( cdot 15^{circ} )
В. ( 60^{circ} )
( c cdot 45^{circ} )
D. ( 30^{circ} )
7
216The incenter of ( Delta A B C ) with vertices
( A(0,0,4), B(3,0,4), C(0,4,4) ) is
A ( .(1,1,1) )
в. (1,1,2)
c. (1,1,3)
D. (1,1,4)
7
217The sides of a right triangle are 9,12 and ( 15 mathrm{cm} ) long. Find the sum of the squares of the medians.
A. ( 327.5 mathrm{cm} )
B . ( 332.5 mathrm{cm} )
c. ( 337.5 mathrm{cm} )
D. ( 322.5 mathrm{cm} )
7
218In an isosceles ( triangle A B C, ) if ( A B=A C )
and ( D ) is a point on ( B C, ) then prove that ( A B^{2}-A D^{2}=B D . C D )
7
219Identify the triangle as equiangular acute, obtuse or right.
Answer: Right angled triangle
Mark answer as 1 if true else 0 if false
( (i) )
7
220The medians of a right triangle which are drawn from the vertices of the acute angles are 5 and ( sqrt{4} 0 . ) The value of the
hypotenuse is:
A . 10
в. ( 2 sqrt{40} )
( c cdot sqrt{13} )
D. ( 2 sqrt{13} )
E. none of these
7
221The angles in a right angled isosceles
triangle are:
A ( cdot 60^{circ}, 60^{circ}, 60^{circ} )
B . ( 90^{circ}, 60^{circ}, 30^{circ} )
c. ( 90^{circ}, 45^{circ}, 45^{circ} )
D . ( 70^{circ}, 50^{circ}, 60^{circ} )
7
222If each side of an equilateral triangle is doubled then its angle will
A. become half
B. be doubled
c. be tripled
D. remain same
7
223In an acute angled triangle ( A B C, A P ) is
the altitude. Circle drawn with AP as its
diameter cuts the sides ( A B ) and ( A C ) at ( D )
and ( mathrm{E} ), respectively, then length DE is equal to
A ( cdot frac{Delta}{2 R} )
в. ( frac{Delta}{3 R} )
c. ( frac{Delta}{4 R} )
D. ( frac{Delta}{R} )
7
224The number of values of b for which
there is an isosceles triangle with sides of lengths ( b+5,3 b-2 ) and ( 6-b ) is/are
A.
B.
( c cdot 2 )
D. 3
7
225( A M ) is a median of a triangle ( A B C . ) Is ( A B+B C+C A>2 A M ? )
(Consider the sides of triangles ( Delta A B M text { and } Delta A M C) )
A. True
B. False
7
226In a triangle the measured of the angles are ( x, x+20, ) and ( 2 x . ) What is the value of
x?
7
227( boldsymbol{E} ) and ( boldsymbol{F} ) are the points on the side ( boldsymbol{P Q} )
and ( P R ) respectively of ( triangle P Q R ) For each
of the following cases, state that ( boldsymbol{E} boldsymbol{F} | )
( Q R ) If True type 1 and for False type 0
( boldsymbol{P E}=mathbf{4} boldsymbol{c m}, boldsymbol{Q} boldsymbol{E}=mathbf{4 . 5} boldsymbol{c m}, boldsymbol{P} boldsymbol{F}= )
( 8 c m ) and ( R F=9 c m )
7
228In the given isosceles triangle, side ( B C=14 c m ) and side ( A B=1 frac{1}{2} l(B C) )
Find the perimeter of the given triangle
7
229In a triangle ( A B C, angle A=90^{circ}, ) and ( A D )
is the altitude. Complete the relation ( frac{B D}{B A}=frac{A B}{D B}-left(frac{ldots ldots .}{A B times B D}right) )
7
230In ( triangle A B C, D ) is a point on ( B C ) such that ( A B=A D=B D=D C, ) then:
( angle A D C: angle C=3: 1 )
A. True
B. False
7
231In a triangle, the largest angle is ten times the smallest and the remaining angle is the square of the smallest. The
largest angle of the triangle is
A ( .80^{circ} )
B . ( 100^{circ} )
( c cdot 120^{circ} )
D. ( 90^{circ} )
7
232It is not possible to construct a triangle when its sides are:
A. ( 8.3 mathrm{cm}, 3.4 mathrm{cm}, 6.1 mathrm{cm} )
B. ( 5.4 mathrm{cm}, 2.3 mathrm{cm}, 3.1 mathrm{cm} )
c. ( 6 mathrm{cm}, 7 mathrm{cm}, 10 mathrm{cm} )
D. 3 cm, 5 cm, 5 cm
7
233In a right angled triangle if an angle
measures then ( 35^{0} ) the measure of
other angle is
A ( .65^{circ} )
B. 55
c. ( 45^{circ} )
D. 30
7
234In the given figure, if AL is the bisector
of ( Delta B A C, ) then ( A B ) is
( A .7 c m )
B. ( 10 mathrm{cm} )
( c .15 c m )
D. ( 22.50 mathrm{cm} )
7
235The base angles of an isosceles triangle
is ( 50^{circ} . ) The size of vertical angle is
A ( .55^{circ} )
B. ( 35^{circ} )
( c cdot 70^{circ} )
D. ( 80^{circ} )
7
236In the given fig ( A D ) divides ( angle B A C ) in
the ratio 1: 3 and ( A D=D B )
Determine the value of ( x )
7
237If two medians of a triangle are equal in length, then the new triangle formed as a result of the medians is:
A. right angled but not isosceles
B. isosceles but not right angle
c. right angled isosceles
D. equilateral
7
238Measure of angle ( angle A C D ) in the given
figure is:
A ( .130^{circ} )
B. 120
( c cdot 150 )
D. ( 115^{circ} )
7
239Suppose that the lines which bisect the
exterior angles at ( B ) and ( C ) of ( Delta A B C )
meet at ( D . ) Then find ( angle B D C )
7
240Find the altitude of an equilateral triangle of side ( 5 sqrt{3} c m )
A ( .7 .5 mathrm{cm} )
в. ( 12.5 mathrm{cm} )
( mathrm{c} .9 .5 mathrm{cm} )
D. ( 8.5 mathrm{cm} )
7
241The lengths of two sides of a triangle are ( 3 mathrm{cm} ) and ( 4 mathrm{cm} . ) Which of the following, can be the length of third side to form a triangle?
A . ( 0.5 mathrm{cm} )
B. ( 5 mathrm{cm} )
( c .8 mathrm{cm} )
D. ( 10 mathrm{cm} )
7
242( ln ) a ( Delta P Q R, ) if ( P Q=P R ) and ( angle Q ) is
twice that of ( angle P, ) then ( angle Q= )
A .72
B. 36
c. ( 144^{circ} )
D. ( 108^{circ} )
7
243Find ( x, ) if the angles of a triangle have
measures ( left(x+40^{0}right),left(2 x+20^{0}right) ) and ( 3 x )
also state which type of triangle is this.
A . ( x=60^{circ} ) and Equilateral triangle
B. ( x=20^{circ} ) and Scalene triangle
C . ( x=20^{circ} ) and Equilateral triangle
D. ( x=40^{circ} ) and Scalene triangle
7
244( A B C D ) is a quadrilateral
( mathbf{s} boldsymbol{A B}+boldsymbol{B C}+boldsymbol{C D}+boldsymbol{D A}>boldsymbol{A C}+boldsymbol{B D} )
A. True
3. Falss
7
245in the given figure if DEllAC and DCIIAP
then which of the
following is ture
A ( cdot operatorname{BE}(mathrm{AD}+mathrm{CP})=mathrm{BE}^{2} )
B. ( mathrm{BE} times mathrm{CP}=mathrm{EC} times mathrm{BC} )
c. ( mathrm{BC} times mathrm{CP}=mathrm{EC} times mathrm{BC} )
D. ( B D times D E times=B E^{2} )
7
246Which is the greatest side in the following triangle? ( angle boldsymbol{A}: angle boldsymbol{B}: angle boldsymbol{C}=mathbf{4}: mathbf{5}: mathbf{6} )
( mathbf{A} cdot A B )
в. ( B C )
( c . A C )
D. cannot be determined
7
247Prove that external angle is sum of the
opposite interior angle.
7
248In the figure find the values of ( x ) and ( y )7
249Using the information, given in each of the following figures, find the valuesof
( a )
( [text { Given }: boldsymbol{C} boldsymbol{E}=boldsymbol{A C}] )
A . ( 136^{circ} )
B. ( 124^{circ} )
( c cdot 110^{circ} )
D. none of the above
7
250An altitude and a median drawn from
the same vertex of a triangle divide the angle at that vertex into.three equal
parts. Prove that the angles of that
triangle are equal to ( 30^{circ}, 60^{circ}, ) and ( 90^{circ} )
7
251In the given triangle ( A B C, ) find the
measure of ( angle boldsymbol{A B C} )
7
252A circle of radius ( 10 mathrm{cm} ) has its centre at the vertex ( C ) of an equilateral triangle ABC and passes through the other two vertices. The side AC extended through C intersects the circle at D. The number
of degrees of angle ADB is:
A ( cdot 15^{circ} )
B . ( 30^{circ} )
( c cdot 60^{circ} )
D. ( 90^{circ} )
E ( .120^{circ} )
7
253( ln a Delta A B C, ) it is given that ( A B=A C )
and the bisectors of ( angle B ) and ( angle C )
intersect at ( 0 . ) If ( mathrm{M} ) is a point on ( mathrm{BO} )
produced, prove that ( angle M O C=angle A B C )
7
254If three angles of a triangle are in the ratio ( 2: 3: 5, ) determine three angles.7
255In an isosceles triangle ( A B C ), with ( A B=A C ), the bisectors of ( angle B ) and ( angle C )
interest each other at ( O ) join ( A ) to ( O )
show that:
( (i) O B=O C )
(ii) ( A O ) bisects ( angle A )
7
256Measure of ( angle A ) in ( Delta A B C ) is
( A cdot 90^{circ} )
В. ( 99.3^{circ} )
( c cdot 100.3^{circ} )
( 800.7^{circ} )
7
257n fig sides QP and RQ of ( Delta ) PQR are produced to the points ( mathrm{S} ) and ( mathrm{T} )
respectively. If ( angle S P R=105^{circ} angle P Q T= )
( 110^{circ} ) then find ( angle P R Q )
( A cdot 35 )
B. ( 70^{circ} )
( c .105 )
0.140
7
258In the figure above, point ( A ) and ( B ) lie on
the circle with center ( O . ) If ( x=80, ) what
is the value of ( z ? )
A . ( 80^{circ} )
B. ( 60^{circ} )
( c cdot 50^{circ} )
D. ( 40^{circ} )
E .10
7
259( P Q R S ) is a square and ( Delta T S R ) is ar
isosceles triangle with ( T S=T R ) Prove
that ( boldsymbol{P T}=boldsymbol{Q} boldsymbol{T} )
7
260In the given figure, ( A M perp B C ) and ( A N )
is the bisector of ( angle A ). Then ( angle M A N ) is
A ( cdot 32 frac{1}{2} )
B . ( 16 frac{1}{2} )
( c cdot 16^{0} )
( D cdot 32^{0} )
7
261In the figure below, AL is perpendicular
to BC and CM is perpendicular to AB. If
( C L=A L=2 B L, ) find ( \$ \$ backslash ) dfrac ( {text { MC }} )
( [mathrm{AM}} )
( A )
B.
( c )
‘. cannot be determine
7
262( A B C ) is a right-angle triangle, right angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are ( 6 mathrm{cm} ) and ( 8 mathrm{cm} ) then radius of the circle is :
( A cdot 3 mathrm{cm} )
B. ( 2 mathrm{cm} )
( c cdot 4 mathrm{cm} )
D. ( 8 mathrm{cm} )
7
263What is the largest angle of the
triangle?
( A cdot x )
B.
( c cdot z )
( D )
7
264Name the following triangle in two different ways.(you mayjudge the nature of the angle by observation)7
265The mean of 10 observations is ( 16.3 . ) If
one observation is registered as 32 instead of 23 , then new mean is
A . 14.8
B. 15.4
( c cdot 16 . )
D. 16.6
7
266Ratio of angle in 5: 3: 1 find the angle.7
267In the given figure, ( angle P R Q=angle P Q R )
then prove that ( angle P Q S=angle P R T )
7
268Show that in a right-angled triangle, the hypotenuse is the longest side.7
269The number of triangles with any three of the length 1,4,6 and ( 8 mathrm{cm} ) as sides is:
A . 4
B. 2
c. 1
( D )
7
270How many medians are there in a
triangle ( P Q R ? )
4
B. 2
( c )
( D )
7
271Two line segments ( A B ) and ( A C ) include an angle of ( 60^{0} ) where ( A B=5 mathrm{cm} ) and ( A C )
( =7 mathrm{cm} . ) Locate points ( mathrm{P} ) and ( mathrm{Q} ) on ( mathrm{AB} )
and ( A C, ) respectively such that ( A P=frac{3}{4} )
( A B ) and ( A Q=frac{1}{4} A C . ) Join ( P ) and ( Q ) and
measure the length PQ.
A. ( 3.25 mathrm{cm} )
B. ( 4.25 mathrm{cm} )
( c .5 .25 mathrm{cm} )
D. ( 6.25 mathrm{cm} )
7
272From the given figure, find ( x )7
273Pis a point inside ( Delta A B C . ) If ( angle P B A= )
( mathbf{2 0}^{circ}, angle boldsymbol{B} boldsymbol{A} boldsymbol{C}=mathbf{5 0}^{circ}, ) and ( angle boldsymbol{P C A}=mathbf{3 5}^{circ} )
then the measure of ( angle B P C ) is
( 4 cdot 65^{circ} )
B. ( 75^{circ} )
( c cdot 90 )
D. 105
7
274Can a triangle have two obtuse angles?
If True enter 1 , else enter 0
7
275What is the number of distinct triangles with integral valued sides and perimeter as ( 14 ? )7
276If ( A, B, C ) are the angles of a triangle, show that ( frac{cos A cos C+cos (A+B) cos (B+C)}{cos A sin C-sin (A+B) cos (B+C)} )
( cot C )
7
277( A B C ) and ( A B D ) are two triangles on
the same base ( A B . ) If line-segment
( C D ) is bisected by ( A B ) at ( O, ) show ( operatorname{that} boldsymbol{a r}(boldsymbol{A B C})=boldsymbol{a r}(boldsymbol{A B D}) )
7
278( mathrm{n} Delta A B C, mathrm{AD} ) bisects ( angle B A C ) and ( mathrm{AD}= )
DC. If ( angle A D B=100^{circ} ), then find ( angle A B D ).
( A cdot 30 )
в. 45
( c cdot 60 )
D. 90
7
279In an equilateral triangle ( A B C, D ) is a point on side ( B C ) such that ( B D=frac{1}{3} )
Prove that ( 9 A D^{2}=7 A B^{2} )
7
280( A B C ) is an isosceles triangle with
vertex at ( A ) and ( P ) is any point inside
the triangle. If the rectangle contained by perpendicular from ( boldsymbol{P} ) to sides ( boldsymbol{A B} ) and ( A C ) is equal to square of the perpendicular from ( boldsymbol{P} ) to base ( boldsymbol{B} boldsymbol{C}, ) then prove that the locus of ( boldsymbol{P} ) is a circle.
7
281State true or false:
Difference of any two sides of a triangle is equal to the third side.
A. True
B. False
7
282If the ratio of the angles of a triangle is
2: 3: 5 then angles are
A ( cdot 36^{circ}, 54^{circ}, 90^{circ} )
В . ( 18^{circ}, 36^{circ}, 126^{circ} )
c. ( 20^{circ}, 60^{circ}, 180^{circ} )
D. ( 18^{circ}, 60^{circ}, 102^{circ} )
7
283Show that ( A(6,4), B(5,-2) ) and ( C(7,-2) ) are
the vertices of an isosceles triangle.
Also, find the length of the median through ( A )
7
284Assertion
Show that the points ( (boldsymbol{a}, boldsymbol{a}),(-boldsymbol{a},-boldsymbol{a}) )
and ( (-sqrt{3} a, sqrt{3} a) ) are the vertices of an
equilateral triangle.
Reason
Using the distance formula we can show that the sides are equal.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
7
285In a triangle ( mathrm{ABC}, angle A=110^{circ} ) and
( A B=A C . ) Find ( angle B ) and ( angle C )
7
286If (-4,0),(0,3),(0,-3) are the vertices
of a triangle, then find the shape of the triangle.
A. isosceles
B. equilateral
c. scalene
D. None of these
7
287The length of vector AG is?
A ( . sqrt{17} )
в. ( frac{sqrt{51}}{3} )
c. ( frac{sqrt{51}}{9} )
D. ( frac{sqrt{59}}{4} )
7
288If the altitudes of a triangle be 3,4,6
then find its in-radius.
7
289From a point 0 in the interior of a
( Delta A B C, ) perpendiculars ( 0 D, O E, & ) OF
are drawn to the sides BC, CA & AB
respectively. Prove that
[i] ( A F^{2}+B D^{2}+C E^{2}=O A^{2}+ )
( O B^{2}+O C^{2}-O D^{2}-O E^{2}-O F^{2} )
[ii] ( A F^{2}+B D^{2}+C E^{2}=A E^{2}+ )
( C D^{2}+B F^{2} )
7
290How do you know the triangle is right
angled?
A. The triangle is right angled when the three sides of a triangle make ( a^{2}+b^{2}=c^{2} )
B. The triangle is right angled when the three sides of a triangle do not make ( a^{2}+b^{2}=c^{2} )
C. The triangle is not right angled when the three sides of a triangle make ( a^{2}+b^{2}=c^{2} )
D. The three sides of a triangle are all same.
7
291In ( Delta A B C, angle A=120, B C+C A=20 )
( A B+B C=21, ) then
( mathbf{A} cdot A B>A C )
B. AB
c. ( triangle A B C ) is isosceles
D. Area of ( Delta A B C=14 sqrt{3} )
7
292n figure, if lines ( mathrm{PQ} ) and ( mathrm{RS} ) intersect at
point ( mathrm{T} ), such that ( angle boldsymbol{P} boldsymbol{R} boldsymbol{T}= )
( 40^{circ}, angle R P T=95^{circ} ) and ( angle T S Q=75^{circ} )
find ( angle S Q T ).
( 4 cdot 20 )
3.60
( c cdot 30 )
( D )
7
293In ( Delta A B C, angle A=43^{circ} ) and ( angle C=70^{circ} )
What is the measure of ( angle B ? )
( mathbf{A} cdot 63^{circ} )
B. ( 65^{circ} )
( c cdot 66^{circ} )
D. ( 67^{circ} )
7
294Prove that in an equilateral triangle, three time the square of a side is equal to four time the square of its altitude.7
295Measure of ( angle B ) and ( angle C ) are
( mathbf{A} cdot 75^{circ}, 85 )
B ( cdot 75^{circ}, 75 )
( mathbf{c} cdot 55^{circ}, 55^{circ} )
D. ( 65^{circ}, 65^{circ} )
7
296Prove that ( angle B O C=90+frac{1}{2} angle B A C )7
297If a ( triangle A B C, ) the bisectors ( angle B ) and ( angle C )
intersect at ( O . ) Prove that ( angle B O C= )
( mathbf{9 0}^{circ}+frac{mathbf{1}}{mathbf{2}} angle boldsymbol{A} )
7
298In
Fig., the length (in ( mathrm{cm} ) ) of each side
has been indicated along the side. State
for each triangle whether it is scalene,
isosceles or equilateral:
7
299In the adjoining figure, ( A B C ) is
triangle in which ( A D ) is the bisector of
( angle A . ) If ( A D perp B C, ) show that ( Delta A B C )
is isosceles.
7
300The side of an equilateral triangle is ( 4 sqrt{3} mathrm{cm} . ) The length of a perpendicular drawn from any vertex to the opposite
side will be
A. ( 4 mathrm{cm} )
B. 6 ст
( c .8 c m )
D. ( 9 mathrm{cm} )
7
301If ( triangle A B C ) has differnence side length
( a, b, c ) and ( a^{2}, b^{2}, c^{2} ) as side will again
form another ( triangle P Q R, ) then ( triangle A B C ) will
always be
A. Acute angle triangle only
B. Obtuse angle triangle only
c. Something Acute or something obtuse depending on value of ( a, b ) and ( c )
D. None of these
7
302Sum of all the three interior angles of a
triangle is equal to
A ( .360^{circ} )
B . ( 120^{circ} )
c. ( 180^{circ} )
D. ( 90^{circ} )
7
30354. I is the incentre of A ABC,
LABC = 60° and ZACB = 50°.
Then Z BIC is :
(1) 55°
(2) 125°
(3) 70°
(4) 65°
7
304Sum of any two sides of a triangle is than the third side.
A. Greater
B. Lesser
c. Equal
D. May be greater or lesser
7
305In ( Delta A B C, ) if ( angle A=50^{circ} ) and ( angle B=60^{circ} )
then the greatest side is :
( A cdot A B )
B. BC
c. АС
D. cannot say
7
306Find the angles ( x ) and ( y ) in the adjoining
figure.
7
307( frac{sin 2 x}{2 cos x}=tan x ? )
A. True
B. False
7
308If the points ( (0,0),(3, sqrt{3}),(p, q) ) form
an equilateral triangle and ( boldsymbol{q}_{1}, boldsymbol{q}_{2} ) are
the two values of ( boldsymbol{q} ) then ( boldsymbol{q}_{1}+boldsymbol{q}_{2}=? )
A ( cdot 2 sqrt{3} )
B. ( sqrt{3} )
( c cdot-sqrt{3} )
D.
7
309| vidnu nas decided to create a triangular
flower bed border. He plans to use 3
pieces of rectangular wooden plates
with lengths 4,5 and 6 feet, as shown in
the figure.

Manu plans to cut the 3 pieces of plate for the flower bed border from a single
piece of plate. If ( frac{1}{8} ) inch wood is wasted
in every cut, calculate the shortest
single piece of plate which can be used
by Manu.
A . 178
B. 179
c. 180
D. 181
E . 182

7
310In the given figure, ( frac{P K}{K S}=frac{P T}{T R} ) and
( angle P K T=angle P R S . ) Prove that ( Delta P S R ) is
an isosceles triangle.
7
311If ( S ) is any point in the interior of ( Delta P Q R ) prove that ( (boldsymbol{S} boldsymbol{Q}+boldsymbol{S} boldsymbol{R})<(boldsymbol{P} boldsymbol{Q}+boldsymbol{P} boldsymbol{R}) )7
312Using triangle inequality theorem check whether the given side lengths
( boldsymbol{a}=mathbf{4}, boldsymbol{b}=mathbf{5} ) and ( boldsymbol{c}=mathbf{8} ) will form a
triangle or not
7
313An exterior angle of a triangle is ( 105^{circ} ) and its two interior opposite angles are equal. Each of these equal angles is
A. ( _{37} frac{1^{0}}{2} )
в. ( _{52} frac{1^{0}}{2} )
c. ( _{72} frac{1^{0}}{2} )
D. 75
7
314In the figure given below, measure of
( angle A B C ) is
A . ( 60^{circ} )
B. 70
( c .80 )
D. 50
7
315If ( G ) is centroid and ( A D, B E, C F ) are three
medians of ABC with area ( 72 mathrm{cm}^{2} ), then
the area of BDG is
A ( cdot 12 ~ c m^{2} )
B. ( 16 mathrm{cm}^{2} )
c. ( 24 mathrm{cm}^{2} )
D. ( 8 mathrm{cm}^{2} )
7
316Area of ( Delta A B C(text { in sq units }) ) is?
A .24
B. ( 8 sqrt{6} )
( c cdot 4 sqrt{6} )
D. None of these
7
317How many isosceles triangles are there
with ( 40^{circ} ) as one of the three angles?
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
7
318Which is the smallest side in the
following triangle? ( angle P: angle Q: angle R=1: 2: 3 )
A. ( P Q )
в. ( Q R )
c. ( P R )
D. cannot be determined
7
319In a circle with center ( 0, ) chords ( A B ) and
CD are of length ( 5 mathrm{cm} ) and ( 6 mathrm{cm} )
respectively and subtend angle ( x^{circ} ) and
( y^{circ} ) at center of circle respectively then
A ( cdot x^{circ}=y^{circ} )
В . ( x^{circ}y^{circ} )
D. None of the above
7
320Which of the following will be the angles of a triangle?
A ( cdot 35^{circ}, 45^{circ}, 90^{circ} )
B . ( 26^{circ}, 58^{circ}, 96^{circ} )
( mathbf{c} cdot 38^{circ}, 56^{circ}, 96^{circ} )
D . ( 30^{circ}, 55^{circ}, 90^{circ} )
7
321Let ( vec{a}, vec{b}, vec{c} ) be position vectors of three vertices of triangle ( A B C, ) find the area of triangle ( A B C )7
322In a ( triangle A B C, ) if ( 2 angle A=3 angle B=6 angle C )
calculate the measures of ( angle B ) (in
degrees)
7
323The angles of triangle are arranged in ascending order of magnitude. If the difference between two consecutive
angles is ( 15^{circ}, ) find the three angles.
7
324( A B C ) is a triangle right-angled at ( B )
and ( D ) is a point on ( B C ) produced
( (B D>B C), ) such that ( B D=2 D C )
Which one of the following is correct?
A. ( A C^{2}=A D^{2}-3 C D^{2} )
B. ( A C^{2}=A D^{2}-2 C D^{2} )
c. ( A C^{2}=A D^{2}-4 C D^{2} )
D. ( A C^{2}=A D^{2}-5 C D^{2} )
7
325The lengths of two sides of a triangle are ( 7 mathrm{cm} ) and ( 10 mathrm{cm} . ) What is the possible
value range of the third side?
A. ( 3 mathrm{cm}< ) third side ( <10 mathrm{cm} )
B. ( 7 mathrm{cm}< ) third side ( <10 mathrm{cm} )
( mathrm{c} cdot 3 mathrm{cm}< ) third side ( <17 mathrm{cm} )
D. ( 7 mathrm{cm}< ) third side
7
326In ( Delta A B C, ) if ( angle A=35^{circ} ) and ( angle B=65^{circ} )
then the longest side of the triangle is :
A . AC
B. AB
c. вс
D. None of these
7
327o Prove: ( -frac{boldsymbol{A} boldsymbol{M}}{boldsymbol{A B}}=frac{boldsymbol{A} boldsymbol{N}}{boldsymbol{A D}} )7
328How many triangles can be drawn
having its angles as ( 45^{0}, 64^{0} ) and ( 72^{0} ? )
( mathbf{A} cdot mathbf{1} )
B . 2
c. More than 2
D. None
7
329The triangle formed by ( B C= )
( 5 c m, A C=3 c m, A B=5.8 c m ) is:
A. a right angled ( Delta )
B. an isosceles ( Delta )
c. an equilateral ( Delta )
D. a scalene ( Delta )
7
330Find the exterior angle.
A . ( 31^{circ} )
B. ( 32^{circ} )
( mathbf{c} cdot 33^{circ} )
D. 34
7
331( A, B, C ) are the three angles of a
triangle ( . ) if ( boldsymbol{A}-boldsymbol{B}=mathbf{1 5}^{circ}, boldsymbol{B}-boldsymbol{C}=mathbf{3 0}^{circ} )
find ( angle boldsymbol{A}, angle boldsymbol{B}, angle boldsymbol{C} )
7
332In the given figure, ABCD is a
parallelogram in which ( angle B D C=45^{circ} )
and ( angle B A D=75^{circ} . ) Then, ( angle C B D=? )
( A cdot 150 )
3. 105
( c cdot 60^{circ} )
( D .75 )
7
333One vertex of the equilateral triangle with centroid at the origin and one side
as ( boldsymbol{x}+boldsymbol{y}-boldsymbol{2}=mathbf{0} ) is?
A ( cdot(-1,-1) )
в. (2,2)
c. (-2,-2)
D. None of these
7
334Fill in the blanks in the following so that each of the following statements is true. ( ln a Delta B C ) if ( angle A=angle C, ) then ( A B=dots )7
335In the given figure (not drawn to scale),
( A B C D ) is a parallelogram, ( A D F ) is an
isosceles triangle with ( boldsymbol{A} boldsymbol{D}=boldsymbol{A} boldsymbol{F} )
( F A B ) and ( E D C ) are straight lines. Find
( boldsymbol{y} )
A . 112
в. 102
( c .132 )
D. 11
7
336( ln Delta A B C, ) if ( A B=B C ) and ( angle B= )
( 80^{circ}, ) then ( angle C= )
A . ( 50^{circ} )
B. 100
( c cdot 130^{circ} )
D. None
7
337n given figure the measure of ( angle A+ )
( angle B+angle C+angle D+angle E+angle F ) is ?
( A cdot 120 )
3.720
( c cdot 360 )
) .540
7
338( ln ) a ( triangle A B C, ) if ( angle A=72^{circ} ) and ( angle B=63^{circ} )
find the ( angle C )
7
339In the adjoining figure, ( A B=A C ) and
( A P perp B C . ) Then:
( mathbf{A} cdot A B=A P )
B. ( A BA P )
D. ( A B leq A P )
7
340Sum of internal angles of a triangle equals to
( mathbf{A} cdot 135^{circ} )
B. ( 180^{circ} )
c. ( 270^{circ} )
D. ( 360^{circ} )
7
341( ln ) a ( Delta P R S, angle P R S=120^{circ} . A ) point ( Q ) is
taken of ( P R ) such that ( P Q=Q S ) and
( Q R=R S ) then ( angle Q P S=dots )
A ( cdot 15^{circ} )
B. ( 30^{circ} )
( c cdot 45^{circ} )
D. ( 12^{circ} )
7
342From the given triangle ( A B C, ) find the
measure of an ( angle A C B )
7
343In ( Delta A B C, angle A=100^{circ}, angle B=30^{circ} ) and
( angle C=50^{circ}, ) then
( mathbf{A} cdot A B>A C )
B. ( A B=A C )
c. ( A B<A C )
D. None of these
7
344Given: ( angle C A B=75^{circ} ) and ( angle C B A=50^{circ} )
Find the value of ( angle D A B+angle A B D )
7
345Find the value of ( y, ) if ( x=5^{circ} )
( A cdot 50 )
3.45
( c cdot 32^{2} )
.38
( ^{2} )
7
346In the figure, the value of ( x )
( A cdot 40 )
B. 70
( c cdot 110^{circ} )
D. 130
7
347If one angle of a triangle equals the sum of the other two angles, the triangle must be
A. scalene
B. right angled
c. obtuse angled
D. acute angled
7
348A triangle can have:
A. one right angle
B. two right angles
C . three obtuse angles
D. none of these
7
349In the figure below, ( angle Q>angle R ) and ( M ) is
a point on ( Q R ) such that ( P M ) is the
bisector of ( angle Q P R ) If the perpendicular
from ( P ) on ( Q R ) meets ( Q R ) at ( N ) then prove that ( angle M P N=frac{1}{2}(angle Q-angle R) )
7
350n given fig. sides ( A B ) and ( A C ) of
( Delta A B C ) are produced to ( E ) and ( D )
respectively. If angle bisectors ( B O ) and
( C O ) of ( angle C B E ) and ( angle B C D ) meet each
other at point ( O, ) then prove that:
( angle B O C=90^{circ}-frac{angle x}{2} )
7
351( mathbf{n} )
Fig., ( mathrm{M}, mathrm{N} ) and ( mathrm{P} ) are the mid-points of
( A B, A C ) and ( B C ) respectively. If ( mathrm{MN}=3 mathrm{cm} ) ( mathrm{NP}=3.5 mathrm{cm} ) and ( mathrm{MP}=2.5 mathrm{cm}, ) calculate
( mathrm{BC}, mathrm{AB} ) and ( mathrm{AC} )
7
352The perimeter of an isosceles triangle is ( 42 mathrm{cm} ) and its base is ( 1 frac{1}{2} ) times its congruent sides. the area of the triangle7
353Construct an isosceles ( Delta A B C ) such
that:
(i) base ( A B=4.2 c m, ) base angle ( = )
( 30^{circ} )
7
354The angles of triangle are in ratio 1: 3:
5 find the angles
( mathbf{A} cdot 20,60,100 )
в. 30,60,90
c. 45,60,75
D. 50,60,70
7
355( ln triangle A B C ) the ( angle B=60^{circ}, angle C=45^{circ} )
Find ( angle A )
7
356Name the types of following triangles:
( triangle A B C ) with ( A B=8.7 mathrm{cm}, A C=7 mathrm{cm} )
and ( B C=6 c m )
7
357In fig ( 9.18, ) tangents PQ and PR are drawn to a circle such that
( angle R P Q=30^{0} . ) A chord RS is drawn
parallel to the tangent PQ. Find the
( angle R Q S ) in degrees.Hint: Draw a line
through ( Q ) and perpendicular to ( Q P )
7
358( boldsymbol{P}(boldsymbol{3}, boldsymbol{4}) boldsymbol{Q}(boldsymbol{7},-boldsymbol{2}) ) and ( boldsymbol{R}(-boldsymbol{2},-1) ) are
vertices of ( triangle P Q R ) write equation of
median
7
359( operatorname{In} ) an acute triangle ( A B C, angle A B C= ) ( mathbf{4 5}^{circ}, boldsymbol{A B}=mathbf{3} ) and ( boldsymbol{A C}=sqrt{mathbf{6}} . ) The angle
( angle B A C, ) is
A ( .60^{circ} )
B. 65
( c cdot 75 )
D. ( 15^{circ} ) or ( 75^{circ} )
7
360If (0,0) and ( (3, sqrt{3}) ) are two vertices of
an equilateral triangle then find third
vertex.
7
361If each angle of a triangle is less than the sum of the other two angles of it then the triangle is right-angled. State
true or false
A. True
B. False
7
362Find the approximate value of ( angle A ) in
( Delta A B C ) if ( 8 angle A=9 angle B=4 angle C )
A ( .40^{circ} )
B. ( 74^{circ} )
( c cdot 86^{circ} )
D. ( 46.3^{circ} )
7
363In ( Delta A B C, ) D and ( E ) are two mid points of
sides ( A B ) and ( A C ) respectively. If
( angle B A C=40^{circ} ) and ( angle A B C=65^{circ}, ) then
( angle C E D ) is
A . 75
B. 125
( c cdot 130 )
D. 105
7
364In an isosceles triangle the sine of the
base angle is three times as large as
the cosine of the vertex angle. Find the sine of the base angle.
7
365In an isosceles triangle ( A B C, ) with
( A B=A C, ) the bisectors of ( angle B ) and ( angle C )
intersect each other ( O . ) Join ( A ) to ( O )
Show that:
( boldsymbol{O} boldsymbol{B}=boldsymbol{O} boldsymbol{C} )
7
366Prove that the median to the base of an
isosceles triangle is perpendicular to
the base.
7
367If two sides of an isosceles ( Delta ) are ( 3 mathrm{cm} )
and ( 8 mathrm{cm}, ) then the length of the third side is :
( mathbf{A} cdot 3 mathrm{cm} )
B. ( 8 mathrm{cm} )
( c .3 mathrm{cm} ) or ( 8 mathrm{cm} )
D. none
7
368The points ( A(2 a, 4 a), B(2 a, 6 a) ) and
( C(2 a+sqrt{3} a, 5 a)(text { when } a>0) ) are
vertices of
A. an obtuse angled triangle
B. an equilateral triangle
c. an isosceles obtuse angled triangle
D. a right angled triangle
7
369Find the coordinates of the vertices of
an equilateral triangle of side 2 a as
shown in the figure.
7
370Name the type of following triangle.
Triangle with lengths of sides ( 7 mathrm{cm}, 8 mathrm{cm} )
and ( 9 mathrm{cm} )
7
371One angle of a triangle is ( 78^{circ} ) and the
other two angles are in the ratio 7: 10 Calculate the bigger angle of the triangle.
7
372n ( triangle A B C, ) the bisector ( A D ) of ( angle A ) is
perpendicular to side ( B C . ) Show that
( A B=A C ) and ( triangle A B C ) is isosceles
7
373PQR is a right-angled triangle with ( mathrm{QS} ) as the perpendicular to the hypotenuse. The ratio PS : SR is equal to
A. ( mathrm{QP}: mathrm{QR} )
B ( cdot Q P^{2}: Q R^{2} )
( c cdot 2: 3 )
D. None of the above
7
374For a triangle ( A B C, ) the true statement is:
A ( cdot A C^{2}=A B^{2}+B C^{2} )
B. ( A C=A B+B C )
c. ( A C>A B+B C )
D. ( A C<A B+B C )
7
375The cosine of the obtuse angle formed by medians drawn from the vertices of
the acute angles of an isosceles rightangled triangle is ( -frac{k}{5}, ) where ( k= )
7
376n fig ( 6.41, ) if ( A B | D E, angle B A C=35^{circ} ) and
( angle C D E=53^{circ}, ) find ( angle D C E )
7
377In the adjoining figure, ( B O ) and ( C O ) are
the bisectors of ( angle B ) and ( angle C ) of ( Delta A B C )
Show that ( angle B O C=90^{circ}+frac{1}{2} angle A )
7
378( angle A D E )7
379In the fig. Straight lines ( A B & C D ) pass
through centre ( O . ) If angle ( angle D C E=40^{circ} )
( & angle A O D=75^{circ} )
Find ( angle C D E & angle O B E )
7
380The angles of triangle are ( 45^{circ}, 60^{circ} ) find the third angle.7
381Find all possible lengths of the third side, if sides of a triangle have 3 and 9
( mathbf{A} cdot 6<x<12 )
B . ( 5<x<12 )
c. ( 6<x<10 )
7
382In the following figure; ( A B ) is the largest
side and ( B C ) is the smallest side of
triangle ( A B C . ) Write the angles ( x^{0}, y^{0} )
and ( z^{0} ) in ascending order of their
values.
A ( cdot y^{0}<z^{0}<x^{0} )
В . ( z^{0}<y^{0}y^{0}>x^{0} )
D. ( z^{0}>x^{0}>y^{0} )
7
383The lengths of the sides of a triangle are proportional to the numbers 5,12 and ( 13 . ) The largest side of the triangle exceeds the smallest side by ( 1.6 mathrm{m} ) Find the perimeter and the area of the triangle.7
384In
( Delta P Q R, ) an exterior angle at ( R ) is
represented by ( 5 x+10 . ) If the two non-
adjacent interior angles are represented by ( 3 x+15 ) and ( 3 x-20 )
find the value of the exterior angle, ( angle R )
A ( .90^{circ} )
B. ( 80^{circ} )
( c cdot 84^{circ} )
D. ( 85^{circ} )
7
385Prove that in an isosceles triangle, the
median to the base is also
perpendicular.
7
386You have studied in Class IX, that a
median of a triangle divides it into two triangles of equal areas. Verify this result for ( triangle A B C ) whose vertices are
( boldsymbol{A}(mathbf{4},-mathbf{6}), boldsymbol{B}(mathbf{3},-mathbf{2}) ) and ( boldsymbol{C}(mathbf{5}, mathbf{2}) )
7
387If two angles in a triangle are ( 75^{circ} ) and 95
then the third angle is ( _{–} )
A. 30
B. 20
( c cdot 10^{circ} )
D. ( 40^{circ} )
7
388Prove that the sum of all interior angles
of a ( Delta ) is ( 180^{circ} )
7
389In an isosceles triangle, one angle has
the angle measure of ( 110^{circ} . ) Find sum of two other angles of the triangle.
A ( .70^{circ} )
B . ( 30^{circ} )
( mathbf{c} cdot 25^{circ} )
D. 26
7
390( triangle A B C ) is an isosceles ( triangle ) with ( A B= )
( A C ) side ( B A ) is produced to ( D ) such that
( A B=A D ) prove ( angle B C D ) is a right
angle.
7
391( ln ) a ( Delta A B C, E ) and ( F ) are the mid-points
of ( A C ) and ( A B ) respectively. The altitude
AP to BC intersects FE at Q. Prove that
( boldsymbol{A} boldsymbol{Q}=boldsymbol{Q} boldsymbol{P} )
7
392Let’s write the measurement of the
angles of triangle ( A B C ) if ( A B=B C )
and ( angle B A C+angle A C B=50^{circ} )
7
393What is the value of ( angle boldsymbol{R} ) ?
A. 100
on
В. ( 110^{circ} )
( c cdot 120 )
0.130
7
394n fig ( 10.49, ) ABCD is parallelogram
CEperpAB,CFperpAD andangleBCE = 40^
Find the values of ( x, y ) and ( z )
7
395In the figure, given below, CE is the
perpendicular to ( A B, angle A C E= )
( 20^{circ} ) and ( angle A B D=50^{circ} . ) Find the
measure of ( angle B D A )
A ( cdot 36^{circ} )
B. ( 60^{circ} )
( c cdot 56^{circ} )
D. none of the above
7
396( ln a Delta A B C, ) if ( A B=A C ) and ( angle A=70^{circ} )
find ( angle B ) and ( angle C )
7
397In each of the following state if the statement is true (T) or false (F):
An equilateral triangle is isosceles also.
A. True
B. False
7
398In ( triangle A B C, angle B ) is a right angle. The median AD and BE are at right angles. Then angle ( C ) is
A . 60
B. ( tan ^{-1}left(frac{1}{sqrt{2}}right) )
c. ( tan ^{-1} sqrt{2} )
D. 90
7
399State true or false:
Sum of two sides of a triangle is greater than the third side.
A. True
B. False
7
400The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive
angles is ( 10^{circ}, ) find the three angles.
7
401In ( Delta A B C, angle B=30^{circ}, angle C=80^{circ} ) and
( angle A=70^{circ} ) then
A ( . A B>B C<A C )
В. ( A BA C )
c. ( A B>B C>A C )
D. ( A B<B C<A C )
7
402If ( triangle A B C ) is right angled at ( A ) and ( A D perp B C, ) then ( frac{B D}{D C}= )
( ^{mathrm{A}} cdotleft(frac{A B}{A C}right)^{2} )
в. ( frac{A B}{A C} )
( ^{mathrm{c}}left(frac{A B}{A D}right)^{2} )
D. ( frac{A B}{A D} )
7
403( A D ) is an altitude of an isosceles
triangle ( A B C ) in which ( A B=A C )
Show that, ( A D ) bisects ( B C )
7
404Which of the following triangle is a type of triangle, classified on the basis of its sides?
A. Acute angle triangle
B. Right angle triangle
c. obtuse angle triangle
D. Equilateral triangle
7
405n the triangle ( A B C, angle A C E=130^{circ} )
( operatorname{segment} A D=A D=D C, ) then find
the measure of an ( angle A B C )
7
406ABC is a triangle and ( A D ) is median. If ( E ) is any point on ( A D, ) then
( mathbf{A} cdot A r(A B E)=A r(A C E) )
В. ( B E=C E )
c. ( A B+B E=A C+C E )
D. ( A E=frac{B E+C E}{2} )
7
407From a point within a triangle segments are drawn to the vertices A necessary
and sufficient condition that the three
triangles formed have equal areas is that the point be
A. such that the three angles formed each have a measure of 120
B. the centre of the inscribed circle
c. the centre of the circumscribed circle
D. the intersection of the medians
7
408The sum of the acute angles of an
obtuse triangle is ( 70^{circ} ) and their
difference is ( 10^{circ} . ) The largest angle is:
A ( cdot 110^{circ} )
B. 105
( c cdot 100^{circ} )
D. ( 95^{circ} )
7
409Calculate to find whether the given
triangle is a right angled triangle.
4.10
B. Yes
C. Insufficient data
D. cannot calculate
7
410Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than the third side.
A . True
B. False
7
411( A triangle A B C ) is right angled at ( A . L ) is a
point on ( B C ) such that ( A L perp B C ) that
( angle B A L=angle A C B )
7
412Mark the correct alternative of the
following.
In figure, the value of ( x ) is?
( A cdot 84 )
B. 74
( c .94 )
D. 57
7
41370. In a triangle ABC, median is AD
and centroid is O. AO = 10 cm.
The length of OD (in cm) is
(1) 6
(2) 4
(3) 5
(4) 3.3
7
414The angles of some triangles are given below. Classify each triangle acuteangled,obtuse-angled or right-angled on the basis of its angles. ( boldsymbol{a} cdot boldsymbol{9} boldsymbol{0}^{o}, boldsymbol{4} boldsymbol{5}^{o}, boldsymbol{4} boldsymbol{5}^{boldsymbol{o}} )
( b .60^{circ}, 60^{circ}, 60^{circ} )
( c .80^{circ}, 60^{circ}, 40^{circ} )
( boldsymbol{d} . boldsymbol{9} boldsymbol{2}^{o}, mathbf{5} boldsymbol{0}^{o}, boldsymbol{3} boldsymbol{8}^{boldsymbol{o}} )
( e .120^{circ}, 50^{circ}, 10^{circ} )
( boldsymbol{f} cdot boldsymbol{9} boldsymbol{0}^{o}, boldsymbol{3} boldsymbol{5}^{o}, boldsymbol{5} boldsymbol{5}^{boldsymbol{o}} )
7
415n figure, show that ( 2(A C+B D)>A B+B C )
( +mathrm{CD}+mathrm{DA} )
7
416State TRUE or FALSE
In a triangle ( A B C, ) right-angled at ( B, B D ) is drawn perpendicular to AC. hence,
( mathrm{CDB}=angle mathrm{A} )
A. True
B. False
7
417( boldsymbol{A}(mathbf{5}, mathbf{4}), boldsymbol{B}(-mathbf{3},-mathbf{2}) ) and ( boldsymbol{C}(mathbf{1},-mathbf{8}) ) are
the vertices of a triangle ABC. Find the equation of median AD.
7
418State whether the statement is
true/false
n a quadrilateral ( A B C D, A B+B C+C D+ )
( mathrm{DA}<2(mathrm{BD}+mathrm{AC}) )
A. True
B. False
7
419In a triangle of base a, the ratio of the other sides is ( r(<1) . ) Show that the
altitude of the triangle is less than or equal to ( frac{a r}{1-r^{2}} )
7
420( ln Delta A B C, B C=A B ) and ( angle B=80^{circ} )
Then ( angle A ) is equal to
A . ( 80^{circ} )
В . ( 40^{circ} )
( c .50^{circ} )
D. ( 100^{circ} )
7
421Triangle ( A B C ) is angled at ( A . A D ) is
drawn perpendicular to ( B C . ) If ( A B= ) ( 10 mathrm{cm}, A C=12 mathrm{cm}, ) find the area of
( triangle A B C . ) Also find the length of ( A D )
7
422In triangle ( A B C, angle A=angle B=52^{circ}, ) write
the name of its largest side.
( A cdot A B )
B. BC
( c cdot c A )
D. None of these
7
423In isosceles triangle ( boldsymbol{A B C}, boldsymbol{A B}=boldsymbol{A C} )
The side ( B A ) is produced to ( D ) such that
( boldsymbol{B A}=boldsymbol{A D} )
Hence, ( angle B C D=90^{circ} )
If the above statement is true then
mention answer as 1 , else mention 0 if
false
7
424If sides of a triangle are ( 5,6, ) and ( 10, ) and the length of the median of biggest side id ( m, ) then find ( 100 m )7
425n figure the sides ( A B ) and ( A C ) of ( Delta A B C )
are produced to point ( mathrm{E} ) and ( mathrm{D} ) repectively. If bisector BO and CO of
( angle C B E ) and ( angle B C D ) respectively meet a point ( 0, ) then ( angle B O C=90^{circ}-frac{1}{2} angle B A C )
7
426If two angles of a triangle are acute angles, then third angle
A. is less than the sum of the two angles
B. is an acute angle
c. is the largest angle of the triangle
D. may be an obtuse angle
7
427fin ( Delta mathrm{ABC}, angle B=90^{circ}, angle C=45^{circ}, ) then
find the angle ( A )
7
428In each of the following figure, find the
value of ( x: )
7
429In the given figure, ( angle boldsymbol{X}= )
( mathbf{6 2}^{circ}, angle boldsymbol{X} boldsymbol{Y} boldsymbol{Z}=mathbf{5 4}^{boldsymbol{o}} . ln triangle boldsymbol{X} boldsymbol{Y} boldsymbol{Z} ) If YO and
ZO are the bisectors of ( angle X Y Z ) and
( angle X Z Y ) respectively find ( angle O Z Y ) and
( angle Y O Z )
7
430If in a ( triangle A B C, cos A=frac{sin B}{2 sin C}, ) then it
is
A. An isosceles triangles
B. An equilateral triangle
C . A right angled triangle
D. None of these
7
431In the figure, measure of ( angle x ) is
( mathbf{A} cdot 65^{circ} )
B ( .85^{circ} )
( c cdot 75 )
D. 55
7
432Show that the sum of the three
altitudes of a triangle is less than the sum of its three sides.
7
433If ( D ) is any point on the side ( B C ) of ( Delta A B C ) such that ( Delta A D B ) and ( Delta A D C )
are equal in area, then
A. AD is the median
B. AD is the altitude
c. AD is an angle bisector
D. AD is any line
7
434In the given figure, find
( angle M N R )
7
435n the given figure, the measure of
( angle B A C ) is
A ( cdot 65^{circ} )
( 3 cdot 50^{circ} )
( c cdot 55^{circ} )
0.60
7
436In the figure given ( Delta A B C ) is a right
isosceles triangle with right angle at ( C )
CD is a parallel to ( A B ) and ( B D=B A ). The
degree measure of ( angle D B C ) equals:
A . ( 10^{circ} )
B . ( 15^{circ} )
( c cdot 20^{circ} )
D. 25
7
437A right angled triangle has a base of ( 40 mathrm{cm}, ) height of ( 30 mathrm{cm} ) and hypotenuse of ( 50 mathrm{cm} . ) The triangle is rotated so that the hypotenuse forms the base. Find the altitude now.7
438In a square ( boldsymbol{P Q R S} ), an equilateral
triangle ( triangle boldsymbol{T} boldsymbol{Q} boldsymbol{R} ) is formed, then
( boldsymbol{m} angle boldsymbol{P T S}-? )
A ( .75^{circ} )
В. ( 90^{circ} )
( c cdot 120^{circ} )
D. ( 150^{circ} )
7
439The length of two sides of triangle are 6
( mathrm{cm} ) and ( 10 mathrm{cm} . ) Between what two wholes
numbers should lie the measure of the
third sides?
7
440( ln ) a ( triangle A B C, A(-2,3), B(2,1) ) and
( C(1,2) . ) What is the foot of the altitude
from the vertex ( A ) of the triangle ( A B C ? )
A ( .(1,4) )
в. (-1,3)
c. (-2,4)
D. (-1,4)
7
441Differences of any two sides of a
triangle is ( ldots ldots . . ) than the third side
A. May be less or more
B. More
c. Equal
D. Less
7
442In figure (ii) given below, ( boldsymbol{A B} | boldsymbol{D} boldsymbol{E} ) and
( boldsymbol{B} boldsymbol{D} | boldsymbol{E} boldsymbol{F} . ) Prove that ( boldsymbol{D} boldsymbol{C}^{2}=boldsymbol{C} boldsymbol{F} times boldsymbol{A} boldsymbol{C} )
7
443The sum of angles of a triangle is equal
to
A ( .90^{circ} )
B . ( 120^{circ} )
( c cdot 150^{circ} )
D. ( 180^{circ} )
7
444The angles of a triangle are ( (2 x+ ) ( left.30^{circ}right),left(3 x-110^{circ}right) ) and ( left(frac{5}{2} x+20^{circ}right), ) find
the value of ( x )
A ( cdot 16^{circ} )
В. ( 32^{circ} )
( c cdot 40^{circ} )
( D cdot 50^{circ} )
7
445In the given triangle the value of ( x ) is
A .55
B. ( 110^{circ} )
( c cdot 60^{circ} )
D. ( 30^{circ} )
7

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