Question

In $△ABC$, the bisector $AD$ of $A$ is perpendicular to side $BC$. Show that $AB=AC$ and $△ABC$ is isosceles.

Answer 1

Given in $\Delta ABC$, the bisector AD of $\angle A$ is perpendicular to side BC.

In $\Delta BAD$ and $\Delta CAD$,

$\angle BAD=\angle CAD$ ....Given

$\angle ADB=\angle ADC={90}^0$ .........(AD is perpendicular to side BC)

$AD=AD$.............(Common side)

$\therefore \Delta BAD\cong \Delta CAD$ ........[ASA test]

$\therefore AB=AC$ ...........[CSCT]

By converse of isosceles triangle theorem,

$\therefore △ABC$ is isosceles. $\left[henceproved\right]$