A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
Circles
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Question

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

KnowledgeBoat · Accepted Answer

From figure, There is a circle with center O and chord AB which is equal to radius of the circle. In ∆ AOB, ⇒ OA = OB = AB ∴ ∆ OAB is an equilateral triangle. ⇒ ∠AOB = 60° (Each angle of an equilateral triangle equals to 90°) We know that, The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. ⇒ ∠AOB = 2∠ACB ⇒ ∠ACB = ∠AOB ⇒ ∠ACB = x 60° = 30° ⇒ ∠ACB = 30°. Since, ACBD is a cyclic quadrilateral ⇒ ∠C + ∠D = 180° (Opposite angles of cyclic quadrilateral are supplementary) ⇒ 30° + ∠D = 180° ⇒ ∠D = 180° - 30° ⇒ ∠D = 150°. Hence, angle subtended on minor arc equals to 150° and angle subtended on major arc equals 30°.

Mathematics

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Circles

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