CBSE Class 10 Maths (Basic) 2025 · Paper 430/6/1 · Question 36

Triangles 4 marks Medium

A triangular window of a building is shown above. Its diagram represents a

△ A B C

with

∠ A = 9 0^{\circ}

and

A B = A C

. Points

P

and

R

trisect

A B

and

P Q ∥ R S ∥ A C

. Based on the above, answer the following questions.

Schematic triangle ABC with trisecting points and parallels

iShow that $△ B P Q \sim △ B A C$ .[1]
iiProve that $P Q = \frac{1}{3} A C$ .[1]
iii(a)If $A B = 3$ m, find length $B Q$ and $B S$ . Verify that $B Q = \frac{1}{2} B S$ .[2]
iii(b)Prove that $B R^{2} + R S^{2} = \frac{4}{9} B C^{2}$ .[2]

Answer

(i)

∠ B

is common and

∠ B P Q = ∠ B A C

(corresponding angles,

P Q ∥ A C

), so

△ B P Q \sim △ B A C

by AA similarity; (ii)

\frac{P Q}{A C} = \frac{B P}{B A} = \frac{1}{3}

, so

P Q = \frac{1}{3} A C

; (iii)(a)

B Q = 2

m and

B S = 22

m, and

\frac{1}{2} B S = 2

= B Q

; (iii)(b)

B R^{2} + R S^{2} = \frac{4}{9} (A B^{2} + A C^{2}) = \frac{4}{9} B C^{2}

. Hence proved.

🔒 Step-by-step solution