In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm.

(a) Prove that △ APB ~ △ CPD

(b) Find the length of CD

(c) Find area △ APB : area △ CPD.

(a) In △ APB and △ CPD,

⇒ ∠APB = ∠CPD (Vertically opposite angles are equal)

⇒ ∠BAP = ∠DCP (Angles in same segment are equal)

∴ △ APB ~ △ CPD (By A.A. axiom)

Hence, proved that △ APB ~ △ CPD.

(b) We know that,

Corresponding sides of similar triangles are proportional.

$\therefore \dfrac{CD}{AB} = \dfrac{PD}{PB} \\[1em] \Rightarrow \dfrac{CD}{9} = \dfrac{2}{3} \\[1em] \Rightarrow CD = 9 \times \dfrac{2}{3} \\[1em] \Rightarrow CD = 6\text{ cm}.$

Hence, CD = 6 cm.

(c) We know that,

Ratio of area of similar triangles is equal to the ratio of square of the corresponding sides.

$\therefore \dfrac{\text{Area of △APB}}{\text{Area of △CPD}} = \dfrac{PB^2}{PD^2} \\[1em] = \dfrac{3^2}{2^2} \\[1em] = \dfrac{9}{4} \\[1em] = 9 : 4.$

Hence, area △ APB : area △ CPD = 9 : 4.