Question

In the figure $2.22$, $M$ is the midpoint of $QR$. $\angle PRQ={90}^o$. Prove that, $P{Q}^2=4P{M}^2-3P{R}^2$

Answer 1

From figure,

$P{Q}^2=P{R}^2+R{Q}^2$

Similarly,

$P{M}^2=P{R}^2+R{M}^2$

$RQ=RM+RQ=RM+RM=2RM$

$\Rightarrow RM=\frac{1}{2}RQ$

$\Rightarrow R{Q}^2=4P{M}^2-4P{R}^2$

$\therefore P{Q}^2=P{R}^2+R{Q}^2$ becomes,

$P{Q}^2=P{R}^2+(4P{M}^2-4P{R}^2)$

$P{Q}^2=4P{M}^2-3P{R}^2$

Hence proved.