Quadrilaterals Questions

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

Quadrilaterals Questions

List of quadrilaterals Questions

Question NoQuestionsClass
1( ln triangle A B C, D ) is the mid-point ( B C . ) If a line is drawn in such a way that it
bisects ( A D ) and ( A D ) and ( A C ) at ( E ) and
( X ) respectively, then prove that ( frac{boldsymbol{E} boldsymbol{X}}{boldsymbol{B} boldsymbol{E}}= )
( frac{1}{3} )
9
2( D ) and ( E ) are the mid-points of the sides
( A B ) and ( A C ) of ( A B C ) and ( O ) is any point
on side ( B C . O ) is joined to ( A . ) If ( P ) and ( Q )
are the mid-points of ( O B ) and ( O C )
respectively, then ( D E Q P ) is:
A . a square
B. a rectangle
c. a rhombus
D. a parallelogram
9
3In a triangle ( A B C, ) a straight line parallel to BC intersects ( A B ) and ( A C ) at points ( D ) and E respectively. If the area of ADE is one fifth of the area of ( mathrm{ABC} ) and ( mathrm{BC}=10 )
( mathrm{cm}, ) then DE equals.
A . ( 2 mathrm{cm} )
B . ( 2 sqrt{5} mathrm{cm} )
( c cdot 4 mathrm{cm} )
D. ( 4 sqrt{5} mathrm{cm} )
9
4Prove the converse of the mid-point theorem following the guidelines given below: Consider a triangle ( A B C ) with ( D ) as the mid-point of ( A B . ) Draw ( D E | B C )
to intersect ( A C ) in ( E . ) Let ( E_{1} ) be the mid-
point of ( A C . ) Use mid-point theorem to
( operatorname{get} D E_{1} | B C ) and ( D E_{1}=B C / 2 )
Conclude ( E=E_{1} ) and hence ( E ) is the
mid-point of ( boldsymbol{A C} )
9
5Suppose the triangle ( A B C ) has an obtuse angle at ( C ) and let ( D ) be the midpoint of side AC Suppose E is on BC such that the segment DE is parallel to AB. Consider the following three statements
i) E is the midpoint of BC
ii) The length of DE is half the length of
( A B )
iii) DE bisects the altitude from ( C ) to ( A B )
A. only (i) is true
B. only (i) and (ii) are true
c. only (i) and (iii) are true
D. all three are true
9
6Prove that a line drawn through the midpoint of one side of a triangle parallel to second side bisects the third side.9
7( 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 ( triangle A B M ) and
( triangle boldsymbol{A} boldsymbol{M} boldsymbol{C} )
9
8A line drawn through the midpoint of one side of a triangle is parallel to another side, so it divides the third side
in the ratio:
A . 1: 1
B . 2: 1
( c cdot 1: 2 )
D. 3: 1
9
9If
( D ) and ( E ) are the midpoint of the sides
( A B ) of a triangle ( A B C, ) Prove that ( boldsymbol{B E}+overline{boldsymbol{D C}}=frac{mathbf{3}}{mathbf{2}} overline{boldsymbol{B} boldsymbol{C}} )
9
1062. In the given figure, ABCD
is a cyclic quadrilateral in which
DC is produced to E and CF is
drawn parallel to AB such that
ZADC = 95°and ZECF =
20°. Then, ZBAD = ?
of
950
F
O
(1) 95°
(3) 105°
Bm
(2) 85°
(4) 75°
f the incircle of a
9
11In the above figure, ( D ) and ( E ) are the
mid-points of the sides ( A C ) and ( B C )
respectively of ( triangle A B C . ) If
( boldsymbol{a} boldsymbol{r}(triangle boldsymbol{B} boldsymbol{E} boldsymbol{D})=12 boldsymbol{c m}^{2}, ) then
( boldsymbol{a} boldsymbol{r}(boldsymbol{A} boldsymbol{B} boldsymbol{E} boldsymbol{D})= )
( mathbf{A} cdot 36 mathrm{cm}^{2} )
B. ( 48 mathrm{cm}^{2} )
( mathrm{c} cdot 24 mathrm{cm}^{2} )
D. None of these
9
12In the quadrilateral (1) given below,
( A B|| D C, E ) and ( F ) are mid point of ( A D )
and ( B D ) respectively. Prove that
( G ) is mid point of ( B C )
9
13In the quadrilateral (1) given below, ( A B|| D C, E ) and ( F ) are mid point of ( A D )
and ( B D ) respectively. Prove that ( boldsymbol{E G}=frac{1}{2}(boldsymbol{A B}+boldsymbol{D C}) )
9
14State true or false:
In the figure, given below, ( 2 A D=A B )
( P ) is mid-point of ( A B . Q ) is mid-point of
DR and ( boldsymbol{P R} | boldsymbol{B S}, ) then ( boldsymbol{A Q} | boldsymbol{B S} )
A. True
B. False
9
15f ( mathrm{D} ) is the mid point of ( mathrm{CA} ) in triangle ( A B C ) and ( Delta ) is the area of triangle, then show that ( tan (angle A D B)=frac{4 Delta}{a^{2}-c^{2}} )9
16In triangle ( A B C, D ) and ( E ) are mid-points of sides ( A B ) and ( B C ) respectively. Also, ( F )
is a point in side ( A C ) so that DF is parallel to BC. Prove that DBEF is a parallelogram. Answer: DBEF is a parallelogram, If true then enter 1 else if False enter 0
9
17( P ) and ( Q ) are the mid points of the sides
( C A ) and ( C B ) respectively of a triangle
( A B C, ) right angled at ( C . ) Then, find the
value of ( 4left(A Q^{2}+B P^{2}right) )
( A cdot 5 A B^{2} )
B . ( 3 A B^{2} ).
c. ( 2 A B^{2} )
( mathrm{D} cdot 4 A B^{2} )
9
1868. ABCD is a parallelogram in which
diagonals AC and BD intersect
at O. If E, F, G and H are the
mid points of AO, DO, CO and
BO respectively, then the ratio
of the perimeter of the quadri-
lateral EFGH to the perimeter of
parallelogram ABCD is
(1) 1:4 (2) 2:3
(3) 1 : 2 (4) 1:3
9
19A cross section at the midpoint of the middle piece of a human sperm will
show
A. Centriole, mitochondria and ( 9+2 ) arrangement of microtubules
B. Centriole and mitochondria
c. Mitochondria and ( 9+2 ) arrangement of microtubules
D. 9+2 arrangement of microtubules only
9
20State true or false:
In ( triangle A B C, A D ) is the median and ( D E )
is parallel to ( B A ), where ( E ) is a point in ( A C ) and Hence ( B E ) is Parallel to BC
A. True
B. False
9
21( ln triangle P Q R, angle P ) is right angle. ( A, B ) are
mid points of ( boldsymbol{R} boldsymbol{P} ) and ( boldsymbol{R} boldsymbol{Q} )
( boldsymbol{R} boldsymbol{P}=boldsymbol{3}, boldsymbol{R} boldsymbol{Q}=mathbf{5} . ) Find length of ( boldsymbol{A} boldsymbol{B} )
4
3
( c .3 )
( D )
9
22In a rectangle ( A B C D, P, Q, R ) and ( S )
are the midpoints of the sides ( A B, B C )
( C D ) and ( D A ) respectively. Find the area
of ( P Q R S ) in terms of area of ( A B C D )
9
23( A B C ) is a triangle right angled at ( C . A ) line through the mid-point ( M ) of
hypotenuse ( A B ) and parallel to ( B C )
intersects ( A C ) and ( D . ) Show that
(i) ( D ) is the mid-point of ( A C )
(ii) ( M B perp A C )
(iii) ( boldsymbol{C M}=boldsymbol{M} boldsymbol{A}=frac{mathbf{1}}{mathbf{2}} boldsymbol{A} boldsymbol{B} )
9
24A triangle ( A B C ) in which ( A B=A C, M ) is a point on ( A B ) and ( N ) is a point on ( A C ) such that if BM=CN then AM=AN
A. True
B. False
9
2569. In AABC, ZC is an obtuse angle.
The bisectors of the exterior an-
gles at A and B meet BC and AC
produced at D and E respective-
ly. If AB=AD = BE, then ZACB =
(1) 1050 (2) 108°
(3) 110° (4) 135°
9
26In figure, ( D E F ) is an isosceles triangle
and ( D G ) is the altitude on ( F E ) is ( D G )
also a median.
9
27Select the correct option given below:
n ( triangle A B C, D, E ) and ( F ) are midpoint of ( A B, B C ) and ( A C ) respectively. If
perimeter of ( triangle A B C=40 ) then find the
perimeter of ( triangle D E F )
4.10
в. ( frac{40}{3} )
( c cdot 20 )
D.
9
28( M ) and ( N ) are midpoints of ( A B ) and ( A C )
respectively ( boldsymbol{A} boldsymbol{M}=mathbf{3} mathbf{c m}, boldsymbol{A} boldsymbol{N}=mathbf{5} ) cm
and ( M N=7 ) cm. Find ( B C )
A. ( 14 mathrm{cm} )
B. ( 10 mathrm{cm} )
( c cdot 6 mathrm{cm} )
D. ( 7 mathrm{cm} )
9
29In a triangle ( triangle A B C, ) points ( P, Q ) and ( R ) are the mid-points of the sides ( A B, B C ) and CA respectively. If the area of the triangle ( A B C ) is 20 sq.units, then area of the triangle PQR equal to:
A . 10 sq. units
B. ( 5 sqrt{3} ) sq. units
c. 5 sq. units
D. 5.5 sq. units
9
3070. In a rectangle ABCD, a point o
is taken such that OA = 5 cm,
OB = 11 cm, OC -12 cm, then
find the value of OD, if the sides
of rectangle are 6 cm and 10 cm.
(1) 6 cm (2) 18 cm
(3) 3/4 cm (4) 413 cm
9
31In triangle ( A B C, M ) is mid-point of ( A B )
and a straight line through ( boldsymbol{M} )
and parallel to ( B C ) cuts ( A C ) in ( N ). Find
the lenghts of ( A N ) and ( M N ) if ( B C=7 )
( mathrm{cm} ) and ( boldsymbol{A C}=mathbf{5} mathrm{cm} )
A. ( A N=2.5 mathrm{cm} ) and ( M N=3.5 mathrm{cm} )
B. ( A N=1.5 mathrm{cm} ) and ( M N=3.5 mathrm{cm} )
c. ( A N=2.5 mathrm{cm} ) and ( M N=4.5 mathrm{cm} )
D. none of the above
9
32If ( D, E ) are the mid-points of ( A B, A C ) of ( triangle A B C ) and ( overline{D E}=lambda overline{B C} ) then ( lambda= )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot frac{1}{2} )
D. 3
9
33n ( Delta P Q R ) are the mid points of side ( A B )
BC and AC respectively. PQ and QR are
joined. Prove that BRQP is a
parallelogram
9
34In the adjoining figure, if ( D ) is the
( tan x )
midpoint of ( mathrm{BC} ), then the value of ( frac{t}{tan y^{0}} )
is:
( A )
B. 3
( c )
D.
9
35In the fig; we have ( X ) and ( Y ) are the mid-
points of ( A C ) and ( B C ) and ( A X=C Y )
then
This question has multiple correct options
A. ( A C=B C )
B. ( A B=B C )
( mathbf{c} . A C=A B )
( mathbf{D} cdot C X=C Y )
9
36In above fig. ( 2, ) if ( D ) is mid-point of ( B C ) the value of ( frac{cot y^{circ}}{cot x^{circ}} ) is:
( A )
B. ( frac{1}{4} )
( c cdot frac{1}{3} )
D. 1
9
37Perpendiculars dropped from the midpoints of two sides of a triangle to the third side are equal.9
38In the given figure, ( A B C ) is a triangle in
which ( mathrm{D} ) and ( mathrm{E} ) are the middle points of
( mathrm{BC} ) and ( mathrm{AC} ) respectively. If ( boldsymbol{A O}=mathbf{6} mathrm{cm} )
find the length of 0 D.
( A cdot 3 mathrm{cm} )
B. ( 6 mathrm{cm} )
( c .4 mathrm{cm} )
( D, 2 mathrm{cm} )
9
39If the line segment joining the midpoint of the consecutive side of quadrilateral ABCD form a rectangle then ABCD must
be
A. Rhombus
B. Square
c. Kite
D. All of the above
9
4057. A point o in the interior of
rectangle ABCD is joined with
each of the vertices A, B, C and
D. Then :
(1) OB + OD = OC + OA
(2) OB2 + OA2 = OC2 + OD2
(3) OB . OD = OC . OA
(4) OB2 + ODP = OC2 + OA?
9
41( ln a Delta A B C, D, E, F ) are respectively
the mid- points of ( B C, C A, ) and ( A B . ) If
the lengths of side ( A B, B C ) and ( C A ) are ( mathbf{7} c boldsymbol{m}, mathbf{8} boldsymbol{c m}, mathbf{9} boldsymbol{c m}, ) respectively, find the
perimeter of ( Delta ) DEF.
A . ( 12 mathrm{cm} )
B. ( 21 mathrm{cm} )
c. ( 24 mathrm{cm} )
D. ( 16 mathrm{cm} )
9
42( M ) is the midpoint of side ( Q R ) of
( triangle P Q R . P S perp Q R ) Prove that: ( P Q^{2}+ )
( boldsymbol{P R}^{2}=mathbf{2 P M}^{2}+mathbf{2 Q} boldsymbol{M}^{2} )
9
43Find the midpoint of the segment connecting the points ( (a,-b) ) and
( (5 a, 7 b) )
A ( cdot(3 a,-3 b) )
в. ( (2 a,-3 b) )
c. ( (3 a,-4 b) )
D. ( (-2 a, 4 b) )
E. none of these
9
44If ( D, E, F ) are respectively the midpoints of the sides ( A B, B C, C A ) of ( Delta A B C ) and the area of ( Delta A B C ) is
( 24 s q . c m, ) then the area of ( Delta D E F ) is:
( mathbf{A} cdot 24 mathrm{cm}^{2} )
B. ( 12 mathrm{cm}^{2} )
c. ( 8 mathrm{cm}^{2} )
D. ( 6 mathrm{cm}^{2} )
9
45State true or false:
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 in
side ( A C ) such that ( E F ) is parallel to ( A B )
Then, ( B E F D ) is a parallelogram.
A. True
B. False
9
46State true or false:
In triangle ( A B C, P ) is the mid-point of
side ( B C . ) A line through ( P ) and Parallel to ( C A ) meets ( A B ) at point ( Q ; ) and a line through ( Q ) and parallel to ( B C ) meets
median ( A P ) at point ( R ). Can it be
concluded that,
( boldsymbol{A P}=boldsymbol{2} boldsymbol{A} boldsymbol{R} ? )
A. True
B. False
9
47ABCD is a quadrilateral in which ( A D= )
BC. If ( P, Q, R, S ) be the mid-points of ( A B )
( A C, C D ) and ( B D ) respectively, show that PQRS is a rhombus.
9
48( X, Y ) are mid-points of opposite sides
( A B ) and ( D C ) of a parallelogram
( A B C D . A Y ) and ( D X ) are joined
ntersecting in ( mathrm{P} ; C X ) and ( B Y ) are
joined intersecting in Q. What is ( boldsymbol{P X Q Y} ) ?
A. Rectangle
B. Rhombus
C. Parallelogram
Dauare
9
49In the quadrilateral (3) given below,
( A B|| D C, E ) and ( F ) are mid point of of
non-parallel sides ( A D ) and ( B C ) respectively. Calculate ( boldsymbol{E F} ) if ( boldsymbol{A B}=boldsymbol{6 c m} ) and ( boldsymbol{D C}=boldsymbol{4 c m} )
9
50Prove that if the mid-points of the opposite sides of a quadrilateral are joined, they bisect each other.9
51In the quadrilateral (2) given below, ( boldsymbol{A B}|boldsymbol{D C}| boldsymbol{E G}, ) If ( boldsymbol{E} ) is mid point of ( boldsymbol{A} boldsymbol{D} )
respectively. Prove that
( 2 E G=A B+C D )
(2)
9
52( D ) and ( F ) are mid-points of sides ( A B )
and ( A C ) of a triangle ( A B C . ) A line
through ( F ) and parallel to ( A B ) meets
( B C ) at point ( E . ) Then ( B D F E ) is a :
A. rhombus
B. parallelogram
c. rectangle
D. data insufficient
9
532NDMISU
6.
Sol.
Find the measure of the angle between the internal bisectors of any two adjacent angles of a parallelogram.
140D
9
54LIDOCOLU
L OOHOU
PULLOUSELL
ABCD is a rhombus with ZABC = 56°, find the value of LACD where AC is the diagonal of rhombus ABCD.
7.
Sol.
ABCD is a rhombus.
9
55D is any point on AC in ( A B C . ) Now,
( boldsymbol{P}, boldsymbol{Q}, boldsymbol{X}, boldsymbol{Y} ) are the mid points of ( mathbf{A B}, mathbf{B C} )
AD and DC respectively. Show that ( P X= )
( mathrm{QY} )
9
56n the adjoining figure ABC is a triangle
in which ( mathrm{D} ) and ( mathrm{E} ) are the mid points ( mathrm{AB} )
and ( A C ) respectively if ( B C=4.6 mathrm{cm} ) then
( D E= )
A. ( 9.2 mathrm{cm} )
B. 2.3 ( mathrm{cm} )
( c .5 .0 mathrm{cm} )
D. ( 6.4 mathrm{cm} )
9
57In given figure, ( P ) and ( Q ) are points on
the sides ( A B ) and ( A C ) respectively of
( triangle A B C ) such that ( A P=3.5 mathrm{cm}, P B= )
( mathbf{7} boldsymbol{c m}, boldsymbol{A} boldsymbol{Q}=mathbf{3} boldsymbol{c m} ) and ( boldsymbol{Q} boldsymbol{C}=boldsymbol{6} boldsymbol{c m} )
( boldsymbol{P Q}=mathbf{4 . 5} boldsymbol{c m}, ) find ( boldsymbol{B C} )
9
58( Delta A B C ) is right angle at ( A(text { figure }) . A D ) is
perpendicular to BC. if ( boldsymbol{A B}= )
( 5 c m, B C=13 c m, A C=12 c m, ) Find
the area of ( Delta A B C . ) Also find the length
of AD.
9
59In a triangle ( A B C, D ) and ( E ) are the
midpoints of ( A B ) and ( A C . ) If the area of
( triangle A B C=60 mathrm{sq.cm}, ) then the area of the
( triangle A D E ) is equal to
A . 15 sq.m
в. 20 sq.m
( c .25 mathrm{sq.m} )
D. 30 sq.m
9
60In triangle ( A B C, X Y | A C ) and divide the triangle into two parts of equal areas. Find the ratio ( frac{boldsymbol{A} boldsymbol{X}}{boldsymbol{A B}} )9
61In ( triangle A B C, D ) is mid-point of ( A B, ) and ( E ) is
mid-point of BC. Calculate : ( D E ), if
( boldsymbol{A C}=mathbf{6 . 4} mathrm{cm} )
A. ( 3.2 mathrm{cm} )
B. ( 3.3 mathrm{cm} )
( c .4 .2 mathrm{cm} )
D. ( 2.6 mathrm{cm} )
9
62( M ) and ( N ) are the midpoints of the diagonals ( A C ) and ( B D ) respectively of quadrilateral ( boldsymbol{A B C D} ), then ( overline{boldsymbol{A B}}+ ) ( A D+overline{C B}+overline{C D}= )
( mathbf{A} cdot 2 overline{M N} )
в. ( 2 overline{N M} )
( c .4 overline{M N} )
D. ( 4 overline{N M} )
9
63rpendiculars drawn on the diagonal AC as shown in the figure. Prove that DN=BM
10. In a rectangle ABCD, DN and BM ar
D
N-
AAN and ACMB we have
9
64n given figure, ( A B C ) is a right triangle
right-angled at B. AD and CE are the two
medians drawn from A and C
respectively. IF ( boldsymbol{A C}=mathbf{5} c boldsymbol{m} ) and ( boldsymbol{A} boldsymbol{D}= )
( frac{3 sqrt{5}}{2} c m, ) find the length of ( C E )
9
65( ln triangle A B C, angle C ) is right angle. D ( & E ) are
mid points of ( A C & A B ) of ( A B=10 & A C= )
6cm. Find length of DE.
A ( .4 mathrm{cm} )
B. ( 5 mathrm{cm} )
( c .3 c m )
D. ( 6 mathrm{cm} )
9
6665. The diagonals AC and BD of a
parallelogram ABCD intersect
each other at the point O such
that ZDAC = 30° and ZAOB =
70°. Then, ZDBC = ?
(1) 40°
(3) 45°
(2) 35°
(4) 50°
ide 6
9
67In figure the segment ( S T ) is parallel to
side ( P R ) of ( triangle P Q R ) and it divides the
triangle into two parts of equal area.Find the ratio ( frac{boldsymbol{P S}}{boldsymbol{P Q}} )
( mathbf{A} cdot 1: 2 )
B. 1: 8
( c cdot 1: 4 )
D. 1: 10
9
68The mid-point of the sides ( A B ) and ( A C ) of ( triangle A B C ) are (3,5) and (-3,-3)
respectively. What is the length of the side BC?
A . 10
B . 20
c. 15
D. 30
9
69( ln triangle P Q R, ) point ( S ) is the midpoint of side
( Q R ). If ( P Q=17, P R=11, P S=13 . ) Find
( Q R )
9
70If ( Delta A B C ) is an isosceles triangle and
midpoints ( D, E, ) and ( F ) of ( A B, B C, ) and
( C A ) respectively are joined, then ( Delta D E F ) is:
A. Equilateral
B. Isosceles
c. Scalene
D. Right-angled
9
7168. Given an equilateral AABC, D, E
and F are the mid-points of AB,
BC and AC respectively. Then the
quadrilateral BEFD is exactly a:
(1) rhombus (2) square
(3) rectangle (4) trapezium
9
72In the given figure, ( ln Delta P Q R quad X Y | )
( Q R, P X: Q X=1: 3 Y R=4.5 mathrm{cm} ) and ( Q R=9 )
( mathrm{cm} . ) Find ( mathrm{XY} ). Further if ( operatorname{ar}(Delta P X Y)=A )
( c m^{2}, ) find in terms of ( A ) the area of
( Delta P Q R ) and the area of trapezium
XYRQ.
9
73( P ) and ( Q ) are mid point of sides RS ( & R T )
respectively of ( triangle ) RST.
( mathrm{PQ}=5 mathrm{cm}, mathrm{RS}=mathrm{RT}=10 mathrm{cm}, ) then
perimeter of ( triangle mathrm{RST} ) is:
A . ( 27.5 mathrm{cm} )
B. ( 20 mathrm{cm} )
( c .25 mathrm{cm} )
D. ( 30 mathrm{cm} )
9
74( ln Delta A B C, ) ray BD bisects ( angle A B C ) and ray
( mathrm{CE} )
bisects ( angle A C B ) If ( operatorname{seg} A B cong operatorname{seg} A C )
then prove that ( E D | B C )
9
75( mathrm{E}, mathrm{F}, mathrm{G} ) and ( mathrm{H} ) are the midpoints of
quadrilateral ABCDas shown in the
figure. If ( mathrm{BD}=8 mathrm{cm}, ) then EH is:
( A cdot 6 c m )
B. 4 cm
( c .5 mathrm{cm} )
D. cannot be found
9
76Prove that the line segment joining the
mid points of the sides of a triangle form four triangles, each of which is similar to the original triangle.
9
7756. In the adjoining figure, sides AB
and AC of a A ABC are produced
to P and respectively. The bi-
sectors of ZPBC and Z9CB in-
tersect at O. Then ZBOC is equal
to :
(1) 90° –
ZBAC
(2) – (ZPBC + Z9CB)
(3) 90° + — ZBAC
(4) None of these
9
78Mid-points of the sides ( A B ) and ( A C ) of ( a ) ( triangle A B C ) are (3,5) and (-3,-3) respectively then the length of the side BC is:
A . 10
B . 20
c. 15
D. 30
9
79Zr= ZAUB = 90
sure of the angles of a quadrilateral ABCD are respectively in the ratio 1:2:3:4. Find the
e respectively in the ratio 1:2:3:4. Find the type of quadrilateral ABCD
9
80(0)
2
57. A point o in the interior of
rectangle ABCD is joined with
each of the vertices A, B, C and
D. Then :
(1) OB + OD = OC + OA
(2) OB2 + OA = OC2 + OD 2
(3) OB. OD = OC . OA
(4) OB2 + OD2 = OC2 + OA?
9
81n ( triangle P Q R, ) seg ( P M ) is the median
Bisectors of ( angle boldsymbol{P} M Q ) and ( angle boldsymbol{P} M boldsymbol{R} )
intersect side ( P Q ) and side ( P R ) in
points ( X ) and ( Y ) respectively, then prove
that ( boldsymbol{X} boldsymbol{Y} | boldsymbol{Q} boldsymbol{R} )
9
8253. The base of a right prison is a quad.
rilateral ABCD. Given that AB = 9
cm, BC = 14 cm, CD = 13 cm, DA
= 12 cm and ZDAB = 90°. If the
volume of the prism be 2070 cm
then the area of the lateral surface
(1) 720 cm (2) 810 cm?
(3) 1260 cm² (4) 2070 cm
9
83In the given figure, if ( E ) and ( F ) are the
midpoints of ( A B ) and ( C D ) of
parallelogram ( A B C D, ) which one is
true?
A ( . C E ) trisects ( B D )
B. ( A F ) trisects ( B D )
c. ( Delta A D F cong Delta C B E )
D. All of these
9
84( A B C ) and BDF are two equilateral triangles such that ( mathrm{D} ) is the mid-point of BC The ratio of the areas of triangles ( A B C ) and BDF is
A . 2: 1
B. 1: 2
c. 4: 1
D. 1: 4
9
85State and prove mid point theorem.9
86In the adjoining figure, ( D ) and ( E ) are
respectively the midpoints of sides ( A B ) and ( A C ) of ( triangle A B C . ) If ( P Q | B C ) and
( C D P ) and ( B E Q ) are straight lines then prove that ( boldsymbol{a} boldsymbol{r}(triangle boldsymbol{A} boldsymbol{B} boldsymbol{Q})=boldsymbol{a} boldsymbol{r}(triangle boldsymbol{A} boldsymbol{C} boldsymbol{P}) )
9
87In the given figure, ( A B C ) is an equilatera
triangle whose side is ( 2 sqrt{3} mathrm{cm} . ) A circle
is drawn which passes through the
midpoints ( D, E ) and ( F ) of its sides. The area of the shaded region is:
( A )
B ( cdot frac{1}{4}(2 pi-sqrt{3}) c m^{2} )
c. ( frac{1}{4}(pi-3 sqrt{3}) c m^{2} )
D.
9
88( A B C D ) is a trapezium in which ( A B | )
( D C, B D ) is a diagonal and ( E ) is the
mid-point of ( A D . A ) line is drawn
through ( E ) parallel to ( A B ) intersecting ( B C ) at ( F ) (see Fig). Show that ( F ) is the
mid-point of ( boldsymbol{B C} )
9
89In a ( triangle A B C, ) let ( P ) and ( Q ) are points on
( A B ) and ( A C ) respectively sucvh that
( P Q mid B C . ) Prove that the median ( A D )
bisects ( boldsymbol{P} boldsymbol{Q} )
9
903.
In the below given figure of parallelogram, find the measure of x°.
DE F
EC
601
609
9
91In triangle ( A B C, P ) is the mid-point of side BC.A line through P and parallel to CA meets ( A B ) at point ( Q ; ) and a line through ( Q ) meets at ( S . Q S ) parallel to BC
meets median AP at point R. prove that:
(i) ( A P=2 A R )
(ii) ( mathrm{BC}=4 mathrm{QR} )
9
9269. What will be the second diag-
onal of a rhombus if the side
of rhombus is 10 cm and one
diagonal is 16 cm?
(1) 6 cm (2) 8 cm
(3) 12 cm (4) 10 cm
9
93n ( Delta A B C ) DE ( | B C ) Find ( A D ) if ( D B= )
( 7.2 mathrm{cm}, A E=1.8 mathrm{cm}, ) and ( E C=5.4 )
( mathrm{cm} )
9
94Tangents PA and PB drawn to ( x^{2}+y^{2}= )
9 from any arbitrary point ‘P’ on the line ( x+y=25 . ) Locus of midpoint of chord
AB is
A ( cdot 25left(x^{2}+y^{2}right)=9(x+y) )
B . ( 25left(x^{2}+y^{2}right)=3(x+y) )
C. ( 5left(x^{2}+y^{2}right)=3(x+y) )
D. None of these
9
95D, ( E ) and ( F ) are the mid-point of sides ( B C )
CA and ( A B ) of ( triangle A B C, ) then area ( triangle A B C )
area ( triangle mathrm{DEF}= )
A . 4: 1
B. 1: 4
c. 2: 1
D. 4: 3
9
96Points ( A, B, C ) and ( D ) are midpoints of
the sides of square ( J ) ETS. If the area of
JETS is 36 sq.cm, then the area of
( A B C D ) is
A. 3 sq. cm
B. 7.5 sq. ( mathrm{cm} )
c. 9 sq. cm
D. 18 sq. cm
9

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