Maharashtra SSC Geometry HOTS Questions with Answers – Similarity | Class 10 Board Exam 2027

Asked in: July 2025 board paper

If \(\Delta ABC \sim \Delta DEF\), \(m\angle B = 60^\circ\), then \(m\angle E\) = ____

Asked in: July 2024 board paper

In \(\Delta ABC\) and \(\Delta PQR\), in a one to one correspondence of vertices, \(\dfrac{AB}{QR} = \dfrac{BC}{PR} = \dfrac{CA}{PQ}\), then ____ statement is true.

[Figure: a small triangle ABC with A at the top left, B below it and C to the right of B; a larger similarly shaped triangle with Q at the top, R below Q and P to the right of R, so that A corresponds to Q, B to R and C to P.]

Asked in: July 2025 and March 2026 board papers (same idea with \(AB : PQ = 2 : 3\) in March 2024)

The ratio of corresponding sides of similar triangles is 3 : 5, then find the ratio of their areas.

Asked in: June 2026 board paper (same idea as a 2-mark activity in July 2025)

In the following figure, seg AE \(\perp\) seg BC, seg DF \(\perp\) line BC, AE = 4, DF = 6, then find \(\dfrac{A(\Delta ABC)}{A(\Delta DBC)}\).

[Figure: \(\Delta ABC\) and \(\Delta DBC\) stand on the same base BC; AE and DF are the perpendiculars from A and D to BC.]

Asked in: July 2024 and June 2026 board papers (same type with LM = 6, MN = 10, TN = 8 in July 2025)

In \(\Delta PQR\), seg RS bisects \(\angle PRQ\). If PR = 15, RQ = 20, PS = 12, then find SQ.

[Figure: \(\Delta PQR\) with point S on side PQ; seg RS bisects \(\angle R\).]

Asked in: March 2025 board paper

In \(\Delta ABC\), DE || BC. If DB = 5.4 cm, AD = 1.8 cm, EC = 7.2 cm, then find AE.

[Figure: \(\Delta ABC\) with D on AB and E on AC, seg DE parallel to side BC.]

Asked in: March 2025 and March 2026 board papers

In the following figure, seg XY || seg AC. If 2AX = 3BX and XY = 9, complete the activity to find the value of AC.

[Figure: \(\Delta ABC\) with X on side AB and Y on side BC, seg XY parallel to side AC.]

Activity: 2AX = 3BX (given) \(\therefore \dfrac{AX}{BX} = \dfrac{3}{\text{____}}\); \(\dfrac{AX + BX}{BX} = \dfrac{3 + 2}{2}\) (by componendo); \(\dfrac{\text{____}}{BX} = \dfrac{5}{2}\) ...... (I). Now \(\Delta BCA \sim \Delta BYX\) (____ test of similarity) \(\therefore \dfrac{BA}{BX} = \dfrac{AC}{XY}\) (corresponding sides of similar triangles) \(\therefore \dfrac{\text{____}}{\text{____}} = \dfrac{AC}{9}\) [from (I)] \(\therefore AC = \) ____

Asked in: July 2024 board paper

In the given figure, AB || CD || EF. If AC = 12, CE = 9, BD = 8, then complete the following activity to find DF.

[Figure: three parallel lines AB, CD, EF cut by two transversals; one transversal passes through A, C, E and the other through B, D, F.]

Activity: AB || CD || EF \(\therefore \dfrac{AC}{\text{____}} = \dfrac{\text{____}}{DF}\) (property of three parallel lines and their transversals) \(\therefore \dfrac{12}{\text{____}} = \dfrac{\text{____}}{DF}\) \(\therefore DF = \dfrac{8 \times 9}{\text{____}}\) \(\therefore DF = \) ____

Asked in: March 2024 and June 2026 board papers

In \(\Delta ABC\), ray BD bisects \(\angle ABC\), A – D – C, seg DE || side BC, A – E – B. Then, to show \(\dfrac{AB}{BC} = \dfrac{AE}{EB}\), complete the following activity.

[Figure: \(\Delta ABC\) with D on AC so that BD bisects \(\angle B\), and E on AB so that DE is parallel to BC.]

Activity: In \(\Delta ABC\), ray BD bisects \(\angle B\). \(\therefore \dfrac{\text{____}}{BC} = \dfrac{AD}{DC}\) ...... (I) (____). In \(\Delta ABC\), DE || BC \(\therefore \dfrac{\text{____}}{EB} = \dfrac{AD}{DC}\) ...... (II) (____). \(\therefore \dfrac{AB}{\text{____}} = \dfrac{\text{____}}{EB}\) [from (I) and (II)]

Asked in: March 2024 board paper (the basic proportionality theorem is the most important proof of this chapter)

Prove that, ‘If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion’.

Asked in: July 2024 board paper

If \(\Delta PQR \sim \Delta LMN\), \(9 \times A(\Delta PQR) = 16 \times A(\Delta LMN)\) and QR = 20, then find MN.

Asked in: March 2025 board paper

In the following figure DE || BC, then: (i) If DE = 4 cm, BC = 8 cm, \(A(\Delta ADE) = 25\) cm\(^2\), find \(A(\Delta ABC)\). (ii) If DE : BC = 3 : 5, then find \(A(\Delta ADE) : A(\square DBCE)\).

[Figure: \(\Delta ABC\) with D on AB and E on AC, seg DE parallel to BC; DBCE is a trapezium.]

Asked in: July 2025 board paper

\(\Delta ABC\) has sides of length 4 cm, 5 cm and 6 cm while \(\Delta PQR\) has perimeter of 90 cm. If \(\Delta ABC\) is similar to \(\Delta PQR\), then find the length of corresponding sides of \(\Delta PQR\).

Asked in: March 2026 board paper

In \(\Delta ABC\), A – X – B and A – Y – C such that segment XY || side BC. Segment XY divides \(\Delta ABC\) in two equal areas, then find \(\dfrac{BX}{AB}\).

If \(\Delta ABC \sim \Delta PQR\) and \(\dfrac{A(\Delta ABC)}{A(\Delta PQR)} = \dfrac{16}{81}\), then AB : PQ = ?

In \(\Delta XYZ\), ray YM bisects \(\angle XYZ\), X – M – Z. If XY = 8 and YZ = 12, then XM : MZ = ?

Two triangles have equal heights. Their areas are 36 cm\(^2\) and 60 cm\(^2\). The ratio of their bases is

In \(\Delta PQR\), seg ST || side QR, P – S – Q and P – T – R. If PS = 3, SQ = 6 and PT = 4, find TR.

At a certain time of day, a vertical pole 6 m high casts a shadow 4 m long. At the same time, how long is the shadow of a vertical tower 9 m high?

\(\Delta ABC \sim \Delta PQR\), \(A(\Delta ABC) = 80\) cm\(^2\), \(A(\Delta PQR) = 125\) cm\(^2\) and AB = 8 cm. Find PQ.

\(\square ABCD\) is a trapezium in which side AB || side DC. Its diagonals AC and BD intersect at O. If AO = 6, OC = 9 and BO = 8, find OD.

[Figure: trapezium ABCD with AB parallel to DC; diagonals AC and BD cross at O.]

In \(\Delta ABC\), point D lies on side BC such that BD : DC = 2 : 3. Find (i) \(A(\Delta ABD) : A(\Delta ADC)\) (ii) \(A(\Delta ABD) : A(\Delta ABC)\).

In \(\Delta ABC\), seg DE || side BC with A – D – B and A – E – C. If AD = x, DB = x − 2, AE = x + 2 and EC = x − 1, find x.

Prove that, ‘The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides’.

In \(\Delta ABC\), seg AD bisects \(\angle BAC\) and D lies on side BC. If AB = 10 cm, AC = 14 cm and BC = 6 cm, find BD and DC.

Points D, E and F are the midpoints of sides AB, BC and CA of \(\Delta ABC\). Show that \(\Delta DEF \sim \Delta CAB\) and find \(A(\Delta DEF) : A(\Delta ABC)\).

A street lamp is fixed on top of a vertical pole 6 m high. A boy 1.5 m tall stands 9 m away from the foot of the pole on level ground. (i) Find the length of his shadow. (ii) How far from the pole must he stand so that his shadow is 4 m long?

In \(\Delta ABC\), A – X – B and A – Y – C with seg XY || side BC. If \(A(\Delta AXY) : A(\square XBCY) = 1 : 8\), find AX : XB.