Question

In the adjoining figure, PA ⊥ AB, QB ⊥ AB and PA = QB. If PQ intersects AB at O, show that O is the midpoint of AB as well as that of PQ.

Answer 1

$$\begin{gathered} \mathrm{In}△\mathrm{AOP}\mathrm{and}△\mathrm{BOQ},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{AOP}=\angle \mathrm{BOQ}(\mathrm{Vertically}\mathrm{opposite}\mathrm{angle}) \\ \angle \mathrm{OAP}=\angle \mathrm{OBQ}(90°\mathrm{each}) \\ \mathrm{PA}=\mathrm{QB}\left(\mathrm{Given}\right) \\ △\mathrm{AOP}\cong △\mathrm{BOQ}(\mathrm{AAS}\mathrm{criterion}) \end{gathered}$$
$$\begin{gathered} \Rightarrow \mathrm{OA}=\mathrm{OB}\left(\mathrm{CPCT}\right) \\ \Rightarrow OP=OQ\left(\mathrm{CPCT}\right) \end{gathered}$$
So, O is midpoint of AB and PQ.