Question

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

Answer 1

Find the length of given line segments

Given: $\angle A=\angle C$, $AB=6cm$, $BP=15cm$, $AP=12cm$ and $CP=4cm$

In$∆APB$and $∆CPD$

$\angle A=\angle C$

$\angle APB=\angle CPD\left[\mathrm{Vertically}\mathrm{opposite}\mathrm{angle}\right]$

Therefore by $\mathrm{AA}$ similarity criteria,

$∆APD~∆CPD$

By basic proportionality theorem,

$$\begin{gathered} \frac{AP}{CP}=\frac{PB}{PD}=\frac{AB}{CD} \\ \Rightarrow \frac{12}{4}=\frac{15}{PD}=\frac{6}{CD} \end{gathered}$$

Considering $\frac{AP}{CP}=\frac{PB}{PD}$ we get;

$$\begin{gathered} PD=\frac{15\times 4}{12} \\ =5cm \end{gathered}$$

Considering $\frac{AP}{CP}=\frac{AB}{CD}$ we get,

$$\begin{gathered} CD=\frac{6\times 4}{12} \\ =2cm \end{gathered}$$

Hence length of $CD=2cm$ and length of $PD=5cm$