Question

two chords AB CD of a circle with center O are produced to meet at P. If angle OPA = angle OPC,prove that AB=CD.

Answer 1

hope this helps you

Answer 2

Proved below.

Step-by-step explanation:

Given:

Here, two chords AB CD of a circle with center O are produced to meet at P

Construction: Draw OP and OQ perpendicular on AB and CD respectively.

⇒ OG = OH  [Equal chords of a circle are equidistant from the center]

Now, in ΔOGP and ΔOHP, we have

OG = OH

OP = OP   [common]

and ∠OGP = ∠OQP = 90°  [By construction]

⇒ ΔOPE ≅ ΔOQE  [RHS congruency]

⇒ GP = HP  [c.p.c.t]                          [1]

⇒ AB = CD          [ As B and D lies on equal sides GP and HP resp[ectively from eq 1]

[Hence proved]