Question

In figure, if ∠A = ∠C, AB = 6 cm, BP = 15 cm, AP = 12 cm and CP = 4 cm, then find the lengths of PD and CD.

Answer 1

According to the question,

∠A = ∠C,

AB = 6 cm,

BP = 15 cm,

AP = 12 cm

CP = 4 cm

From ∆APB and ∆CPD,

∠A = ∠C

∠APB = ∠CPD  ...[Vertically opposite angles]

∴ By AAA similarity criteria,

ΔAPD ∼ ΔCPD

Using basic proportionality theorem,

⇒ `("AP")/("CP") = ("PB")/("PD") = ("AB")/("CD")`  ...[By basic proportionality theorem]

⇒ `12/4 = 15/("PD") = 6/("CD")`

Considering `("AP")/("CP") = ("PB")/("PD")`, we get,

`12/4 = 15/("PD")`

PD = `(15 xx 4)/12`

= `60/12`

= 5 cm

Considering, `("AP")/("CP") = ("AB")/("CD")`

⇒ CD = `((6 xx 4))/12` = 2 cm

Therefore,

Length of PD = 5 cm

Length of CD = 2 cm