Question

In an isosceles ∆ABC, the base AB is produced both the ways to P and Q such that AP ✕ BQ = AC². Prove that ∆APC ∼ ∆BCQ.

Answer 1

It is given that is isosceles and _.

We have to prove that .

It is given that is an isosceles triangle, so AC = BC.

Now,

(Given)

$\Rightarrow \frac{AP}{AC}=\frac{BC}{BQ}$
Also,

$$\begin{gathered} \angle CAB=\angle CBA\left(\mathrm{Equal}\mathrm{sides}\mathrm{have}\mathrm{equal}\mathrm{angles}\mathrm{opposite}\mathrm{to}\mathrm{them}\right) \\ \Rightarrow 180°-\angle CAP=180°-\angle CBQ \\ \Rightarrow \angle CAP=\angle CBQ \end{gathered}$$

Hence, (SAS Similarity)