Question

In an isosceles $\Delta ABC$, the base AB is produced both the ways to P and Q such that $AP\times BQ=A{C}^2$. Prove that $\Delta APC\sim \Delta BCQ$.

Answer 1

We need to prove that ΔAPC∼ΔBCQ

Given ΔABC is an isosceles triangle AC = BC.

Now, AP x BQ = AC² (given)

AP x BQ = AC x AC

Also, ∠CAB = ∠CBA (equal sides have angles equal opposite to them)

180 – ∠CAP = 180 – ∠CBQ [ Linear pair]

∠CAP = ∠CBQ

Hence, ΔAPC∼ΔBCQ (SAS similarity)