Question

$\text{In the given figure, OA = OB and OP = OQ.}\text{Prove that (i) PX = QX, (ii) AX = BX.}$

Answer 1

In △OQA and △OPB, we have:

OQ=OP (Given)

OA=OB (Given)

∠AOQ=∠BOP

△OQA ≅ △OPB

∠OAQ=∠OBP (Corresponding angles of congruent triangles)

Now, consider triangles BQX and APX.

Given: OA=OB
OP=OQ

∴ OA-OP=OB-OQ

⇒AP=BQ

Further, ∠BXQ=∠AXP (Vertically opposite angles)

Also, we have proven that ∠QBX=∠PAX.

∆BQX≅∆APX (AAS criterion)

∴ PX=QX (corresponding sides of congruent triangles)

Also, AX=BX (corresponding sides of congruent triangles)

Hence, proved.