Question

In figure, AP and BQ are perpendiculars to the line segment AB and $AP=BQ$, Prove that O is the midpoint of the line segment AB as well as PQ.

Answer 1

In $△OAP$ and $△OBQ$,

$AP=BQ$(given)

$\angle OAP=\angle OBQ={90}^{\circ }$

$\angle OAP=\angle OBQ$(vertically opposite angles)

$\therefore △OAP$ is congruent to $△OBQ$ by AAS axiom

$\therefore OA=OB$ by C.P.C.T.

and $OP=OQ$ by C.P.C.T

$\Rightarrow O$ is the midpoint of line segments $AB$ and $PQ$