Question

In the given figure, PA ⊥ AB, QB ⊥ AB and PA = QB.
Prove that ∆OAP ≅ ∆OBQ.
Is OA = OB?

Figure

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ PA\perp AB \\ QB\perp AB \\ PA=QB \\ \mathrm{To}\mathrm{prove}:△OAP\cong △OBQ \\ \mathrm{Find}\mathrm{whether}OA=OB. \\ \mathrm{Proof}: \\ \mathrm{In}△OAP\mathrm{and}△OBQ: \\ \angle POA=\angle QOB(v\mathrm{ertically}\mathrm{opposite}\mathrm{angles}) \\ \angle OAP=\angle OBQ(90°\mathrm{each}) \\ PA=QB\left(g\mathrm{iven}\right) \\ ByAAS\mathrm{congruence}\mathrm{property}: \\ △OAP\cong △OBQ \\ =>OA=OB(\mathrm{corresponding}\mathrm{parts}\mathrm{of}\mathrm{the}\mathrm{congruent}\mathrm{triangles}) \end{gathered}$$