Chapter-wise questions
Triangles
1. If the areas of two similar triangles are in the ratio 16 : 49, write the ratio
of their corresponding sides.
2. In the given figure, ∠APC = 90º, AB = 12cm, BC = 13cm, AP = 3cm and
PC = 4cm. Find ∠BAC
A
C
P
B
3. P and Q are the points on the sides CA and CB respectively of ∆ABC,
right angled at C. Prove that AQ² + BP² = AB² + PQ²
C
F
D
E
A
B
Q
4.
In the given figure, PQ = 24cm, QR = 26cm, ∠PAR = 90, PA = 6cm and AR = 8cm,
Find ∠QPR
P
A
º
90
P
R
5.
In the given figure, If PQ = 12cm, PX = 4cm, PY = 3cm, YR = 6cm and XY = 5cm.
Find QR
X
Q
Y
R
Chapter-wise questions
P
6.
In the given figure AB ║ QR. And AB : QR = 2 : 3
find ar(∆PAB) : ar (∆PQR)
7.
If ∆ABC ~ ∆QPR, ∠A = 32º and ∠R = 48º. Find ∠B
8.
The perpendicular from vertex A on the side BC of the triangle ABC intersect BC at
A
B
R
Q
point D, such that BD = 3CD. Prove that 2AB² = 2AC² + BC²
9.
In a ∆ABC, D and E are points on the sides AB and AC respectively and DE║BC.
Find x if AD = 4cm, AE = 8cm, DB = x–4 and EC = 3x – 19.
10. The perimeter of two similar triagles are 36 cm and 18 cm respectively. If one side
of the first triangle is 18 cm, find the corresponding side of the other.
R
11. Find the unknown measures
x and y for the given similar triangles
16
A
C
51º
x
y
8
P
6
20
39º
Q
B
12. P, Q, and R are the midpoints of the sides, AB, BC and AC respectively of ∆ABC.
find ar(PQR) : ar (ABC)
A
1
13. In the figure PQ is parallel to BC and AP = 1cm,
PB = 3cm what is the ratio of the area of
∆APQ to area of ∆ABC?
P
Q
3
B
C
14. In two right triangles ∆ABC and ∆PQR, right angled at B & R,
∠A = 41º and ∠Q = 49º.
Prove that AB =
PQ x CA
PR
15. The short and long hand of a clock are 5cm & 12cm respectively. What will be the
distance between their tips at 3’0 clock?
Chapter-wise questions
16. In the given figure A is the center of the circle and BD⊥ AC.
Prove that AB X BD = AD X BC.
B
A
D
C
17. ABC is an isosceles triangle, in which AB = AC, circumscribed about a circle.Show
that BC is bisected at the point of contact.
18. E is a point on the side AD produced of a parallelogram ABCD and BE intersects
CD at F. Show that ∆ABE ~ ∆CFB.
19. ∆ABC and ∆DEF are similar. The area of ABC is 9 sq. cm and area of DEF is 16
sq. cm. If BC = 2.1cm, find the length of EF. 20. In the given figure ABD is a right triangle, right angled at A and AC⊥BD.
Prove that AB² = BC x BD
Prove that
BE
BC
=
.
EC
CP
C
A
21. In the given figure DE║AC and DC║AP,
D
B
A
D
B
E
C
P
22. Any point X inside triangle DEF is joined to its vertices. From a point P in DX, PQ
is drawn parallel to DE meeting XE at Q and QR is drawn parallel to EF meet XF
at R. Prove that PR║DF.
23. In a triangle ABC, DE║BC and If AC = 5.6, find AE.
AD
3
=
. DB
5
A
D
B
E
C
Chapter-wise questions
24. In the given figure, QA and PB are perpendicular to AB. If AO = 10cm,
BO = 6cm and PB = 9cm. Find AQ
P
A
10cm O
9cm
6cm B
Q
25. The perimeters of two similar triangles ABC and PQR are respectively 36 cm and
24cm. If PQ = 10cm, find AB.
K
26. In the given figure, PQ is parallel to MN. If KP
4
=
and KN = 20.4cm. Find KQ PM 13
P
Q
M
N
L
27. In the given figure, P
a
express x in terms of a, b and c.
M
46º
x
b
N
28. D is a point on side BC of ∆ABC such that ∠ADC = ∠BAC. Prove that
46º
c
K
CA
CB
=
CD
CA
29. Two triangles ∆BAC and ∆BDC, right angled at A and D respectively, are drawn on
the same base BC and on the same side of BC. If AC and DB intersect at P, prove
that AP X PC = DP x PB.
30. Any line parallel to the parallel sides of a trapezium divides the nonparallel sides
proportionally. 5 cm
A
B
31. In the given figure
AO
BO
1
= = OC
OD
2
O
and AB = 5cm. Find the value of DC.
D
C
32. Let ABC be a triangle and D and E be two points on side AB such that AD = BE. If
DP║BC and EQ║AC, then prove that PQ║AB
33. The diagonal BD of a parallelogram ABCD intersects the segment AE at the point
F, where E is any point on the side BC. Prove that DF X EF = FB X FA.
34. Prove that the diagonals of a trapezium divide each other proportionally.
Chapter-wise questions
35. In a square PQRS the diagonals PR and QS intersect at T, prove that
PQ = PR X PT
R
36. In the figure given below A is a point on PQ such that AQ:PQ = 3:5. AB is parallel to PR.
B
C
In triangle ABC, ∠C = 90º and in ∆PRD, ∠D = 90º. Calculate the length of PD, if BC = 9cm.
D
P
A
Q
37. O is a point inside an equilateral triangle PQR such that OP, OQ and OR are
the bisectors of ∠P, ∠Q and ∠R respectively. The bisector of ∠POQ, ∠QOR and
∠ROP meet the side PQ, QR and PR at points A, B and C respectively. Show that
PA x QB x RC = PC x QA X RB.
A
38. In the given figure,
1
CD.
3
Prove that 2CA² = 2AB² + BC²
AD⊥BC and BD =
B
D
C
39. P and Q are points on sides AB and AC respectively of ∆ABC. If AP = 3 cm, PB =
6cm, AQ = 5cm, and QC = 10cm, show that BC = 3 PQ
40. Through the mid point M of the side CD of a parallelogram ABCD, the line BM is
drawn intersecting AC in L and AD produced in E. Prove that EL = 2 BL.
41. Two poles of height a meters and b meters are p meters apart. Prove that the
height of the point of intersection of the lines joining the top of each pole to the foot
ab
of the opposite pole is given by
meters. a+b
42. ABC is a triangle in which AB= AC and D is a point on AC such that BC² = AC x CD.
Prove that BD = BC
43. In given figure, S and T trisect the side QR
of a right triangle PQR.
Prove that 8PT² = 3PR² + 5PS². P
Q
........................
S
T
R
Chapter-wise questions
44. If BL and CM are medians of a triangle ABC right-angled at A, then prove that
4(BL² + CM²) = 5BC².
A
45. In the given figure if EF║DC║IAB. Prove that
B
E
AE BF
.
=
ED FC
F
D
C
46. If the diagonals of a quadrilateral divided each other proportionally, prove that it is
a trapezium. 47. In ∆ABC, AD║BC and AD = BD.DC. Prove that BAC is a right angle.
48. In a triangle, if the square of one side is
equal to the sum of the squares of the other
two sides, prove that the angle opposite
to the first side is a right angle. Use the
above theorem to find the measure of ∠PKR in given figure.
8cm
P
m
24c
K
Q
6c
m
R
26cm
49. Prove that the ratio of the areas of two similar triangles is equal to the ratio of
the squares on their corresponding sides. Using the above do the following: The
diagonals of a trapezium ABCD, with AB║CD, intersect each other at the point O.
If AB = 2CD, find the ratio of the area of ∆AOB to the area of ∆COD.
50. Prove that the ratio of the areas of two similar triangles is equal to the ratio of
squares of their corresponding sides.
Using the above result, prove the following:
In a ∆ABC, XY is parallel to BC and it divides ∆ABC into two parts of equal area.
Prove that 51.
BX
√2–1
=
AB
√2
If a line is drawn parallel to one side of a triangle
intersecting the other two sides, then it divides
the two sides in the same ratio. Using the above,
prove the following:
In the fig., AB║DE and BC║EF. Prove that AC║DF. D
A
O
E
B
C
F
Chapter-wise questions
D
52. Prove that in a right triangle, the square of the hypotenuse
is equal to the sum of the squares of the remaining two sides.
Making use of the above, prove the following:
In the figure below, ABCD is a rhombus. Prove that 4AB² = AC²+BD²
O
A
C
B
53. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the
squares of their corresponding sides.
Use the above prove the following: If the areas of two similar triangles are equal,
prove that they are congruent.
54.
A
If a line is drawn parallel to one side of a triangle, to
intersect the other two sides in distinct points,
D
E
prove that the other two sides are divided in the
B
C
same ratio. Using the above,
prove the following: In the given figure DE║BC and BD = EC. Prove that ABC is an
isosceles triangle.
55. State and prove Pythagoras theorem. Use the above to prove the following:
ABC is an isosceles right triangle, right angled at C. Prove that
AB² = 2AC²
56. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the
squares of their corresponding sides.
Using the above, do the following:
The areas of two similar triangles ABC and PQR are in the ratio 9 :16 . If BC = 4.5
cm, find the length of QR.
C
57. If a line is drawn parallel to one side of a triangle, prove
that the other two sides are divided in the same ratio.
Using the above result, prove from the given figure that AD = BE
if and ∠A = ∠B & DE ║ AB. A
D
E
B
58. Prove that the ratio of the areas of two similar triangles is equal to the ratio of
squares of their corresponding sides.
Use the above in the following:
In a trapezium ABCD, O is the point of intersection of AC and BD, ABII CD and
AB = 2 CD. If the area of AOB = 84 cm², find the area of ∆COD.
Chapter-wise questions
A
59. In the given figure, DE║BC and AD:DB = 5:4.
Area(∆DEF)
Find
Area (∆CFB)
D
E
F
B
C
60. In a right-angled triangle the square on the hypotenuse is equal to the sum of
squares on the other two sides. Prove it.
Using the above prove the following:
In ∆ABC, D is the mid-point of BC and AE⊥BC. If AC > AB, show that
1
AB² = AD² – BC.DE +
BC².
4
61. In a triangle, if the square on one side is equal to the sum of the squares on the
remaining two, the angle opposite the first side is a right angle. Prove it.
Using the above prove the following: If in a ∆PQR PS⊥QR and PS² = QS X RS,
then is right – angled at P.