Multiple choice

In a circle with centre O, BC is a chord. Point D and A are on the circle, on the opposite side of BC, such that ∠DBC = 280 and BD = DC. What is the measure of ∠BOC?

  1. 98°

  2. 84°

  3. 112°

  4. 96°
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C Correct answer
Explanation

The angle shown as 280 is evidently 28°. Since BD = DC, triangle BDC is isosceles, so its angle at D is 180° - 28° - 28° = 124°. The corresponding minor central angle is 360° - 2 x 124° = 112°.

AI explanation

In triangle BDC, BD = DC, so it is an isosceles triangle with angle BDC = 180 - 2(28) = 124 degrees. Since angles subtended by chord BC on opposite sides of a circle add to 180 degrees, angle BAC = 180 - 124 = 56 degrees. The central angle theorem states that the central angle is twice the inscribed angle subtended by the same chord, so angle BOC = 2 * 56 = 112 degrees. The measure of angle BOC is 112 degrees.

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