Question

16. In the given figure, AB and CD are two mutually perpendicular
chords of a circle. Prove that the sum of arc BC and arc AD is equal to the
semi circumference of the circe.
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Answer 1

Answer:

Let O be the center of a circle

Join BC, OA, OB, OC and OD

Given: AB⊥ CD

In Δ ABK

∠ ABC + ∠ BCD + ∠ BKC = 180° (Angle sum property)

⇒ ∠ ABC + ∠ BCD + 90° = 180°

⇒ ∠ ABC + ∠ BCD = 90° …..(1)

∠AOC = 2∠ ABC (∵ angle at center twice angle in

segment) ...(2)

∠BOD = 2∠ BCD (∵ angle at center twice angle in

segment) ...(3)

Adding (2) and (3)

∠AOC + ∠BOD = 2∠ ABC + 2∠ BCD

∠AOC + ∠BOD = 2 (∠ ABC + ∠ BCD)

∠AOC + ∠BOD = 2 × 90° (from (1))

∠AOC + ∠BOD = 180°

Hence,