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,

⇒ ${\displaystyle \frac{\text{AP}}{\text{CP}}=\frac{\text{PB}}{\text{PD}}=\frac{\text{AB}}{\text{CD}}}$  ...[By basic proportionality theorem]

⇒ ${\displaystyle \frac{12}{4}=\frac{15}{\text{PD}}=\frac{6}{\text{CD}}}$

Considering ${\displaystyle \frac{\text{AP}}{\text{CP}}=\frac{\text{PB}}{\text{PD}}}$, we get,

${\displaystyle \frac{12}{4}=\frac{15}{\text{PD}}}$

PD = ${\displaystyle \frac{15\times 4}{12}}$

= ${\displaystyle \frac{60}{12}}$

= 5 cm

Considering, ${\displaystyle \frac{\text{AP}}{\text{CP}}=\frac{\text{AB}}{\text{CD}}}$

⇒ CD = ${\displaystyle \frac{(6\times 4)}{12}}$ = 2 cm

Therefore,

Length of PD = 5 cm

Length of CD = 2 cm