Question

In the given figure, $\angle LMN={90}^o$ and $\angle LKN={90}^o$. seg MK $\perp$ seg LN.

Prove that R is the midpoint of seg MK.

Answer 1

Consider $△LMN$

$MR$ perpendicular to $LN$ and $LM$ perpendicular to $LN$

$\therefore$ By property of geometric mean, ${MR}^2=LR.RN$....(i)

Similarly in $△LKN$,

$KR$ perpendicular to $LN$ and $LK$ perpendicular to $KN$

$\Rightarrow {KR}^2=LR.RN$....(ii)

$\therefore$ From (i) and (ii)

${MR}^2={KR}^2\Rightarrow MR=KR$

$\therefore R$ is the midpoint of segment $MK$