Question

State true or false:

In $\Delta ABC,$ angle $ABC$ is equal to twice the angle $ACB$, and the bisector of angle $ABC$ meets
the opposite side at the point $P$. Then, $AB\times BC=BP\times CA$

• A. True
• B. False

Answer 1

The correct option is A True

Given: $\angle ABC=2\angle ACB$, BP bisects $\angle ABC$
$\angle ABC=2\angle ACB$
$\frac{1}{2}\angle ABC=\angle ACB$
$\angle ABP=\angle ACB$ (AP bisects $\angle ACB$) (I)
Now, In $△APB$ and $△ABC$
$\angle ABP=\angle ACB$ (From I)
$\angle BAP=\angle BAC$ (Common)
$\angle APB=\angle ACB$ (third angle)
thus, $△APB\sim △ABC$
Hence, $\frac{AB}{AC}=\frac{BP}{BC}$...(Corresponding sides)
$AB\times BC=BP\times AC$