Question 12 Marks

$\triangle PQR$ is isosceles with $\text{PQ} = \text{PR}. T$ is the mid$-$point of $\text{QR},$ and $\text{TM}$ and $\text{TN}$ are perpendiculars on $\text{PR}$ and $\text{PQ}$ respectively. Prove that,
$\text{PM} = \text{PN}$

Answer

Since$, \triangle QNT ≅ \triangle RMT$
$\text{NQ} = \text{MR} .............(ii)$
$But \text{PQ} = \text{PR} .........(iii) ...($given$)$
Subtracting $(ii)$ from $(iii)$
$\text{PQ} - \text{NQ} = \text{PR} - \text{MR}$
$\Rightarrow \text{PN} = \text{PM}.$

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