Question

In $\Delta ABC$ and $\Delta PQR$, $\angle ABC\cong \angle PQR$, seg $BD$ and seg $QS$ are angle bisector. If $\frac{l\left(AD\right)}{l\left(PS\right)}=\frac{l\left(DC\right)}{l\left(SR\right)}$
Prove that : $\Delta ABC\sim \Delta PQR$

Answer 1

The angle bisector of a triangle dividing one side into two segments.

The ratio of the two segments is proportional to the ratio of the other two sides of the triangle.

According to the problem,

In triangle ABC, BD is the angle bisector.

Then $AD/DC=AB/AC$ ………..$\left(1\right)$

Similarly, in triangle PQR, QS is the angle bisector.

Then $PS/SR=PQ/QR$ ………….$\left(2\right)$

Given that $AD/DS=PQ/QR$

Substitute $AD=(DC.PS)/SR$

On solving, $AB/AC=PS/SR$

Therefore $AB/AC=PQ/SR$

The sides are proportional to each other.

Hence the $\Delta ABC\sim \Delta PQR$.