Question

What is an intercepted arc and inscribed angle?

Prove measure of an intercepted arc is twice the measure of an inscribed angle.

Answer 1

An inscribed angle is an angle formed by two chords in a circle which have a common endpoint.  The other two endpoints that form an arc is called as an intercepted arc on the circle.

• Given:
• O is the centre of a circle and arc AB subtending ∠AOB at the centre and ∠BPA at any point on the axis of the circle.
• To Prove:
• ∠AOB=2∠BPA
• Construction:
• Join PO and produce it to a point Q.
• Proof:
• In ΔAOP,
• OA = OP (∵ radii of a circle)
• ∠OPA=∠OAP (∵ angles opp. the equal sides)
• Also, ∠QOA=∠OPA+∠OAP(∵ exterior angle of a triangle)
• ⇒∠QOA=∠OPA+∠OPA  (∵ ∠OPA=∠OAP)
• ⇒∠QOA=2∠OPA-----------(a)
• Similary, by taking ΔBOP,
• ⇒∠QOB=2∠OPB-----------(b)
• Adding (a) and (b), we get,
• ⇒∠QOA+∠QOB=2∠OPB+2∠OPA=2(∠OPB+∠OPA)
• ⇒∠AOB=2∠APB