Question

State true or false:

In $△ABC$, $\angle A$ is obtuse and $AB=AC$. $P$ is any point in side $BC$. $PM\perp AB$
and $PN\perp AC.$
Then, $PM\times PC=PN\times PB$

• A. True
• B. False

Answer 1

The correct option is A True

Given, $AB=AC$
$\angle ABC=\angle ACB$ (Isosceles triangle property) (I)
In $△PMB$ and $△PNC$
$\angle MBP=\angle NCP$ (From I)
$\angle PMB=\angle PNC$ (each ${90}^{\circ }$)
$\angle MPB=\angle NPC$ (third angle)
Thus, $△PMB\sim △PNC$ (AAA rule)
Hence, $\frac{PM}{PN}=\frac{PB}{PC}$ (Corresponding sides)
Thus $PM\times PC=PB\times PN$