Question

In the given figure, $\frac{PK}{KS}=\frac{PT}{TR}$ and $\angle PKT=\angle PRS$. Prove that $\Delta PSR$ is an isosceles triangle.

Answer 1

Given:

$\frac{PK}{KS}=\frac{PT}{TR}$

$\angle PKT=\angle PRS$

To prove that $\Delta PSR$ is an isosceles triangle

Proof:

$\frac{PK}{KS}=\frac{PT}{TR}$

$\frac{KS}{KP}=\frac{RT}{PT}$

$\frac{KS}{KP}+1=\frac{RT}{PT}+1$

$\frac{KS+KP}{KP}=\frac{RT+PT}{PT}$

$\frac{PS}{KP}=\frac{PR}{PT}$

$\angle KPT$ is common angle to $\Delta KPT$ and $\Delta SPR$

Using SAS similarity of triangles, $\Delta KPT\sim \Delta SPR$

$⟹\angle PKT=\angle PSR$

$⟹\angle PRS=\angle PSR\because \angle PKT=\angle PRS$

Since, two angles of $\Delta PSR$ are equal, it is isosceles triangle.

Hence proved