Triangles

CBSE Class 10 — Maths — Previous Year Questions

83 questions216 total marksVSA: 23MCQ: 24Assertion-Reason: 1SA: 9LA: 26

Q1.VSA1 mark(2020)Easy

△ A BC \sim △ PQR

, if

\frac{A B}{PQ} = \frac{1}{3}

, then find

\frac{ar ( △ A BC )}{ar ( △ PQR )}

Q2.VSA1 mark(2020)Easy

A BC

is an equilateral triangle of side

2 a

, then length of one of its altitude is ________.

Q3.VSA1 mark(2020)Easy

In Fig. 1,

MN ∥ BC

and

A M : MB = 1 : 2

, then

\frac{ar ( △ A MN )}{ar ( △ A BC )} =

________.

$Triangle ABC with MN parallel to BC$

Q4.VSA1 mark(2020)Easy

△ A BC

A B = 63

cm,

A C = 12

cm and

BC = 6

cm, then

∠ B =

________.

Q5.VSA1 mark(2020)Easy

Two triangles are similar if their corresponding sides are ________.

Q6.MCQ1 mark(2020)Easy

△ A BC \sim △ D EF

such that

A B = 1.2

cm and

D E = 1.4

cm, the ratio of the areas of

△ A BC

and

△ D EF

Q7.VSA1 mark(2020)Easy

Let

△ A BC \sim △ D EF

and their areas be respectively 81 cm

^{2}

and 144 cm

^{2}

. If

EF = 24

cm, then length of side

BC

is ________ cm.

Q8.MCQ1 mark(2020)Easy

In Figure 1,

A BC

is an isosceles triangle, right-angled at

C

. Therefore

$Isosceles right triangle ABC with right angle at C$

Q9.MCQ1 mark(2023)Easy

△ A BC \sim △ PQR

with

∠ A = 32°

and

∠ R = 65°

, then the measure of

∠ B

is:
(a)

32°

(b)

65°

(c)

83°

(d)

97°

Q10.MCQ1 mark(2023)Easy

In the given figure,

D E ∥ BC

. If

A D = 2

units,

D B = 3

units and

A E = 3

units and

EC = x

units, then the value of

x

is:

$Triangle with DE parallel to BC$

(a)

2

(b)

3

(c)

5

(d)

\frac{9}{2}

Q11.MCQ1 mark(2023)Easy

In the given figure,

A B ∥ PQ

. If

A B = 6

cm,

PQ = 2

cm and

OB = 3

cm, then the length of

OP

is:
(a)

9

cm
(b)

3

cm
(c)

4

cm
(d)

1

Q12.MCQ1 mark(2023)Easy

In the given figure,

△ A BC \sim △ QPR

. If

A C = 6

cm,

BC = 5

cm,

QR = 3

cm and

PR = x

, then the value of

x

is:

$Similar triangles ABC and QPR$

Q13.MCQ1 mark(2023)Easy

△ A BC

PQ ∥ BC

. If

PB = 6

cm,

A P = 4

cm,

A Q = 8

cm, find the length of AC.

$Triangle ABC with PQ parallel to BC$

Q14.MCQ1 mark(2023)Easy

In the given figure,

D E ∥ BC

. The value of

x

is:

$Triangle with DE parallel to BC$

Q15.MCQ1 mark(2023)Medium

△ A BC

and

△ D EF

\frac{A B}{D E} = \frac{BC}{F D}

. Which of the following makes the two triangles similar?

Q16.MCQ1 mark(2023)Easy

△ PQR \sim △ A BC

;

PQ = 6

cm,

A B = 8

cm and the perimeter of

△ A BC

36

cm, then the perimeter of

△ PQR

Q17.MCQ1 mark(2023)Easy

In the given figure,

D E ∥ BC

. If

A D = 3

cm,

A B = 7

cm and

EC = 3

cm, then the length of

A E

is

$Triangle with DE parallel to BC$

Q18.MCQ1 mark(2024)Easy

In the given figure

△ A BC

is shown.

D E

is parallel to

BC

. If

A D = 5

cm,

D B = 2.5

cm and

BC = 12

cm, then

D E

is equal to

$Triangle ABC with DE parallel to BC$

Q19.MCQ1 mark(2024)Easy

The perimeters of two similar triangles

A BC

and

PQR

are

56

cm and

48

cm respectively.

\frac{PQ}{A B}

is equal to

Q20.MCQ1 mark(2024)Easy

△ A BC

D E ∥ BC

(as shown in the figure). If

A D = 4

cm,

A B = 9

cm and

A C = 13.5

cm, then the length of

EC

is:

$Triangle ABC with DE parallel to BC$

Q21.MCQ1 mark(2024)Easy

In the given figure, in

△ A BC

D E ∥ BC

. If

A D = 2.4

cm,

D B = 4

cm and

A E = 2

cm, then the length of

A C

is:

Q22.MCQ1 mark(2024)Easy

If a vertical pole of height 7.5 m casts a shadow 5 m long on the ground and at the same time, a tower casts a shadow 24 m long, then the height of the tower is:

Q23.Assertion-Reason1 mark(2024)Medium

Assertion (A):

A BC D

is a trapezium with

D C ∥ A B

E

and

F

are points on

A D

and

BC

respectively, such that

EF ∥ A B

. Then

\frac{A E}{E D} = \frac{BF}{FC}

.

Reason (R): Any line parallel to parallel sides of a trapezium divides the non-parallel sides proportionally.

Q24.MCQ1 mark(2025)Easy

In triangles ABC and DEF,

∠ B = ∠ E

∠ F = ∠ C

and

A B = 3 D E

. Then, the two triangles are:

Q25.MCQ1 mark(2025)Easy

In the given figure,

PQ ∥ BC

. If

\frac{A P}{PB} = \frac{4}{13}

and

A C = 20.4

cm, then the length of

A Q

is:

$Triangle ABC with PQ parallel to BC$

Q26.MCQ1 mark(2025)Easy

Which of the following statements is **incorrect**?

Q27.MCQ1 mark(2025)Medium

If in two triangles

△ D EF

and

△ PQR

∠ D = ∠ Q

and

∠ R = ∠ E

, then which of the following is *not* true?

Q28.MCQ1 mark(2025)Medium

The measurements of

△ L MN

and

△ A BC

are shown in the figure given below. The length of side AC is:

$Similar triangles LMN and ABC with measurements$

Q29.MCQ1 mark(2025)Medium

In the given figure, in

△ A BC

A D ⊥ BC

and

∠ B A C = 90°

. If

BC = 16

cm and

D C = 4

cm, then the value of

x

is:

$Triangle ABC with AD perpendicular to BC, BC = 16 cm, DC = 4 cm$

Q30.MCQ1 mark(2025)Medium

△ A BC

and

△ PQR

are shown in the adjoining figures. The measure of

∠ C

is:

$Triangles ABC and PQR with sides AB=3.8cm, AC=3√3cm, BC=6cm and PQ=12cm, PR=6√3cm, QR=7.6cm, angle R=80°, angle Q=60°$

Q31.MCQ1 mark(2025)Medium

E and F are points on the sides AB and AC respectively of a

△ A BC

such that

\frac{A E}{EB} = \frac{A F}{FC} = \frac{1}{2}

. Which of the following relation is true?

Q32.VSA2 marks(2020)Medium

In the given figure,

D E ∥ A C

and

D C ∥ A P

. Prove that

\frac{BE}{EC} = \frac{BC}{CP}

.

$Triangle ABP with DE parallel to AC and DC parallel to AP$

Q33.VSA2 marks(2020)Medium

In Fig. 5,

A BC

and

D BC

are two triangles on the same base

BC

. If

A D

intersects

BC

O

, show that

\frac{ar ( △ A BC )}{ar ( △ D BC )} = \frac{A O}{D O}

$Rhombus ABCD with diagonals$

Q34.VSA2 marks(2020)Medium

In Fig. 6, if

A D ⊥ BC

, then prove that

A B^{2} + C D^{2} = B D^{2} + A C^{2}

.

$Triangle with AD perpendicular to BC$

Q35.VSA2 marks(2020)Easy

The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one side of the first triangle is 9 cm long, find the length of the corresponding side of the second triangle.

Q36.VSA2 marks(2020)Medium

In Figure-5,

△ PQR

is right-angled at

P

M

is a point on

QR

such that

PM

is perpendicular to

QR

. Show that

P Q^{2} = QM \times QR

.

$Triangle PQR right-angled at P, M on QR$

Q37.VSA2 marks(2020)Medium

In the given Figure 5,

D E ∥ A C

and

D F ∥ A E

. Prove that

\frac{BF}{FE} = \frac{BE}{EC}

.

$Triangle ABC with points D on AB, E and F on BC, DE parallel to AC and DF parallel to AE$

Q38.VSA2 marks(2023)Medium

In the given figure,

XZ

is parallel to

BC

A Z = 3

cm,

ZC = 2

cm,

BM = 3

cm and

MC = 5

cm. Find the length of

X Y

.

$Triangle ABC with XZ parallel to BC, points X on AB, Y on XZ, Z on AC$

Q39.VSA2 marks(2023)Easy

In the given figure,

A BC

is a triangle in which

D E ∥ BC

. If

A D = x

D B = x - 2

A E = x + 2

and

EC = x - 1

, then find the value of

x

Q40.VSA2 marks(2023)Medium

Diagonals

A C

and

B D

of trapezium

A BC D

with

A B ∥ D C

intersect each other at point

O

. Show that

\frac{O A}{OC} = \frac{OB}{O D}

.

$Trapezium ABCD with diagonals intersecting at O$

Q41.VSA2 marks(2024)Medium

In the given figure, ABCD is a quadrilateral. Diagonal BD bisects

∠ B

and

∠ D

both. Prove that:
(i)

△ A B D \sim △ CB D

(ii)

A B = BC

$Quadrilateral ABCD with diagonal BD$

Q42.VSA2 marks(2024)Easy

In the given figure,

△ A HK \sim △ A BC

. If

A K = 8

cm,

BC = 3.2

cm and

HK = 6.4

cm, then find the length of

A C

.

$Triangle AHK similar to triangle ABC$

Q43.VSA2 marks(2025)Easy

△ A BC \sim △ PQR

in which

A B = 6

cm,

BC = 4

cm,

A C = 8

cm and

PR = 6

cm, then find the length of

(PQ + QR)

Q44.VSA2 marks(2025)Medium

In the given figure,

\frac{QR}{QS} = \frac{QT}{PR}

and

∠1 = ∠2

. Show that

△ PQS \sim △ TQR

.

$Triangle with points P, Q, R, S, T$

Q45.VSA2 marks(2025)Medium

In the given figure, D is a point on the side BC of

△ A BC

such that

∠ A D C = ∠ B A C

. Show that

C A^{2} = C D \cdot CB

.

$Triangle ABC with point D on BC, angle ADC = angle BAC$

Q46.VSA2 marks(2025)Medium

In the given figure,

O A \cdot OB = OC \cdot O D

. Show that

∠ A = ∠ C

and

∠ B = ∠ D

.

$Quadrilateral with diagonals intersecting at O, OA.OB = OC.OD$

Q47.VSA2 marks(2025)Medium

A 1.5 m tall boy is walking away from the base of a lamp post which is 12 m high, at the speed of 2.5 m/sec. Find the length of his shadow after 3 seconds.

Q48.VSA2 marks(2025)Medium

In parallelogram ABCD, side AD is produced to a point E and BE intersects CD at F. Prove that

△ A BE \sim △ CFB

Q49.SA3 marks(2020)Medium

In the given figure,

∠ D = ∠ E

and

\frac{A D}{D B} = \frac{A E}{EC}

. Prove that

B A C

is an isosceles triangle.

$Triangle ABC with points D on AB and E on AC, Fig. 5$

Q50.SA3 marks(2020)Medium

In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle.

Q51.SA3 marks(2020)Medium

In Fig. 7, if

△ A BC \sim △ D EF

and their sides of lengths (in cm) are marked along them, then find the lengths of sides of each triangle.

$Two similar triangles ABC and DEF with sides marked$

Q52.SA3 marks(2020)Medium

In a right triangle, prove that the square of the hypotenuse is equal to the sum of squares of the other two sides.

Q53.SA3 marks(2020)Medium

Construct a triangle with sides 5 cm, 6 cm and 7 cm. Now construct another triangle whose sides are

\frac{2}{3}

times the corresponding sides of the first triangle.

Q54.SA3 marks(2020)Medium

Construct a

△ A BC

with sides

BC = 6

cm,

A B = 5

cm and

∠ A BC = 60°

. Then construct a triangle whose sides are

\frac{3}{4}

of the corresponding sides of

△ A BC

Q55.SA3 marks(2023)Hard

In the given figure,

C D

is the perpendicular bisector of

A B

EF

is perpendicular to

C D

A E

intersects

C D

G

. Prove that

\frac{CF}{C D} = \frac{FG}{D G}

Q56.SA3 marks(2023)Hard

In the given figure,

A BC D

is a parallelogram.

BE

bisects

C D

M

and intersects

A C

L

. Prove that

E L = 2 B L

.

$Parallelogram ABCD with E, M, L$

Q57.SA3 marks(2023)Medium

In the given figure,

E

is a point on the side

CB

produced of an isosceles triangle

A BC

with

A B = A C

. If

A D ⊥ BC

and

EF ⊥ A C

, then prove that

△ A B D \sim △ ECF

Q58.LA4 marks(2020)Medium

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then construct another triangle whose sides are

\frac{4}{5}

times the corresponding sides of the first triangle.

Q59.LA4 marks(2020)Hard

In an equilateral triangle

A BC

D

is a point on the side

BC

such that

B D = \frac{1}{3} BC

. Prove that

9 A D^{2} = 7 A B^{2}

.

$Equilateral triangle ABC with point D on BC and altitude AE$

Q60.LA4 marks(2020)Medium

Prove that the sum of squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

$Rhombus ABCD with diagonals intersecting at O$

Q61.LA4 marks(2020)Medium

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

Q62.LA4 marks(2020)Medium

In a right-angled triangle, prove that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Q63.LA4 marks(2020)Hard

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Q64.LA5 marks(2023)Medium

If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, prove that the other two sides are divided in the same ratio.

Q65.LA5 marks(2023)Hard

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of

△ PQR

. Show that

△ A BC \sim △ PQR

Q66.LA5 marks(2023)Hard

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

E L = 2 B L

Q67.LA5 marks(2023)Medium

D is a point on the side BC of a triangle ABC such that

∠ A D C = ∠ B A C

, prove that

C A^{2} = CB \cdot C D

Q68.LA5 marks(2023)Hard

If AD and PM are medians of triangles ABC and PQR, respectively where

△ A BC \sim △ PQR

, prove that

\frac{A B}{PQ} = \frac{A D}{PM}

Q69.LA5 marks(2024)Hard

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

Q70.LA5 marks(2024)Hard

In the given figure PA, QB and RC are each perpendicular to AC. If

A P = x

BQ = y

and

CR = z

, then prove that

\frac{1}{x} + \frac{1}{z} = \frac{1}{y}

.

$Triangles with PA, QB, RC perpendicular to AC$

Q71.LA5 marks(2024)Medium

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

Q72.LA5 marks(2024)Hard

Sides

A B

and

A C

and median

A D

△ A BC

are respectively proportional to sides

PQ

and

PR

and median

PM

of another triangle

PQR

. Show that

△ A BC \sim △ PQR

Q73.LA5 marks(2024)Medium

E

is a point on the side

A D

produced of a parallelogram

A BC D

and

BE

intersects

C D

F

. Show that

△ A BE \sim △ CFB

Q74.LA5 marks(2024)Hard

Sides

A B

BC

and the median

A D

△ A BC

are respectively proportional to sides

PQ

QR

and the median

PM

△ PQR

. Prove that

△ A BC \sim △ PQR

Q75.LA5 marks(2024)Hard

In the given figure,

△ FEC ≅ △ G D B

and

∠1 = ∠2

. Prove that

△ A D E \sim △ A BC

.

$Triangle ABC with points D, E on sides, angles 1, 2, 3, 4 marked$

Q76.LA5 marks(2024)Hard

Sides

A B

and

A C

and median

A D

of a

△ A BC

are respectively proportional to sides

PQ

and

PR

and median

PM

of another

△ PQR

. Show that

△ A BC \sim △ PQR

Q77.LA5 marks(2025)Hard

The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that

D F \times EF = FB \times F A

.

$Parallelogram ABCD with diagonal BD and line AE$

Q78.LA5 marks(2025)Hard

△ A BC

, if

A D ⊥ BC

and

A D^{2} = B D \times D C

, then prove that

∠ B A C = 90°

.

$Triangle ABC with AD perpendicular to BC$

Q79.LA5 marks(2025)Hard

Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points divides the other two sides in the same ratio. Hence, in the figure given below, prove that

\frac{A M}{MB} = \frac{A N}{N D}

where

L M ∥ CB

and

L N ∥ C D

.

$Quadrilateral ABCD with LM parallel to CB and LN parallel to CD$

Q80.LA5 marks(2025)Hard

In the given figure, PA, QB and RC are perpendicular to AC. If

P A = x

units,

QB = y

units and

RC = z

units, prove that

\frac{1}{x} + \frac{1}{z} = \frac{1}{y}

.

$PA, QB, RC perpendicular to AC with intersection point Q on AC$

Q81.LA5 marks(2025)Hard

Sides AB and BC and median AD of triangle ABC are respectively proportional to sides PQ and QR and median PM of

△ PQR

. Show that

△ A BC \sim △ PQR

Q82.LA5 marks(2025)Medium

The corresponding sides of

△ A BC

and

△ PQR

are in the ratio

3 : 5

A D ⊥ BC

and

PS ⊥ QR

as shown in the following figures:

(i) Prove that

△ A D C \sim △ PSR

(ii) If

A D = 4

cm, find the length of

PS

.
(iii) Using (ii) find

ar (△ A BC) : ar (△ PQR)

Q83.LA5 marks(2025)Hard

State basic proportionality theorem.
Use it to prove the following:
If three parallel lines

l

m

n

are intersected by transversals

q

and

s

as shown in the adjoining figure, then

\frac{A B}{BC} = \frac{D E}{EF}

.

$Three parallel lines l, m, n intersected by transversals q and s$

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