ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120° and ∠BAC = 30°, the minimum possible value of ∠BCD is _____
Question
ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120° and ∠BAC = 30°, the minimum possible value of ∠BCD is _____
Solution
To solve this problem, we need to use the properties of a cyclic quadrilateral and the circle.
Step 1: In a circle, the angle subtended by an arc at the center is double the angle subtended by it at any point on the alternate segment. So, if ∠COD = 120°, then ∠CAB = 1/2 * ∠COD = 60°.
Step 2: Now, we know that ∠BAC = 30°. Since ∠BAC and ∠CAB are angles on a straight line (they are supplementary), we can find ∠ABC by subtracting ∠BAC from 180°. So, ∠ABC = 180° - ∠BAC - ∠CAB = 180° - 30° - 60° = 90°.
Step 3: In a cyclic quadrilateral, the sum of opposite angles is 180°. So, if ∠ABC = 90°, then ∠BCD = 180° - ∠ABC = 180° - 90° = 90°.
Therefore, the minimum possible value of ∠BCD is 90°.
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