Question

In the given figure , ABC is an isosceles triangle in which AB$=$AC. E is point on the side CB produced such that FE $\perp$ AC . If AD $\perp$ CB. Prove that : AE $\times$ EF $=$ AD $\times$ EC.

Answer 1

REF.image

To prove : $AB\times EF=AD\times EC$

$\Rightarrow \frac{AB}{EC}=\frac{AD}{EF}$

Proof :

AB=AC $(\because$ ABC is isosceles)

$\therefore \angle B=\angle C$ (angles opposite to equal sides are equal) - (1)

In $\Delta ABD$ and $\Delta ECF$

$\angle ABD=\angle ECF$ (from (1))

$\angle ADB=\angle EFC$ (Both are ${90}^{\circ }$)

Using AA similarity

$\Delta ADB\sim \Delta ECF$

$\Rightarrow \frac{AB}{EC}=\frac{AD}{EF}$

$\Rightarrow AB\times EF=AD\times EC$

$\therefore$ Hence proved.