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In the given figure, $PS$ is the bisector of $\angle QPR$ of $∆PQR$. Prove that $\frac{QS}{SR}=\frac{PQ}{PR}$
"
Given:
$PS$ is the bisector of $\angle QPR$ of $∆PQR$.
To do:
We have to prove that $\frac{QS}{SR}=\frac{PQ}{PR}$
Solution:
Draw $RT \| PS$,meeting $QP$ produced at $T$.
$PS \| RT$
This implies,
$\angle 2=\angle 3$ (Alternate angles)
$\angle 1=\angle 4$ (Corresponding angles)
$\angle 1 = \angle 2$ (AD is the bisector of $\angle A$)
Therefore,
$\angle 3 =\angle 4$
This implies,
$PT=PR$
In $\triangle QRT$,
$\frac{QS}{SR}=\frac{PQ}{PT}$
$\frac{QS}{SR}=\frac{PQ}{PR}$
Hence proved.
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