Question

In the figure, $\Delta ABC$ is an isosceles triangle in which AB = AC P, Q and R are the mid points BC, AC and AB respectively. Show that $AP\perp RQ$ and AP is bisected by RO

Answer 1

$\Delta ABC$ is an isosceles triangle with AB = AC. P, Q and R be the mid points of sides BC, CA and AB respectively.
To Prove. $AP\perp RQ$ and AP is bisected by RQ.
Proof. Since the line segment joining the mid points of two sides of a triangle is parallel to third side and half of it,
$\therefore PQ||ABandPQ=\frac{1}{2}AB$
and $PR||ACandPR=\frac{1}{2}AC$

Since AB = AC [Given]
$\therefore$ PQ = PR ...(i)
Also, $AR=\frac{1}{2}AB$ $[\because R$ is the mid point of AB]
and, $AQ=\frac{1}{2}AC$ ...(iii)
From (i), (ii) and (iii), we get
AR = PR = PQ = AQ
$\Rightarrow ARPQ$ is a rhombus.
Since diagonal of a rhombus bisect each other at right angle, therefore
$AP\perp QR$ and AP is bisected by QR.