Mathematics
In the given figure, AD is a diameter of a circle with centre O and AB is tangent at A. C is a point on the circle such that DC produced intersects the tangent at B. If ∠ABC = 50°, find ∠AOC.
Circles
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Answer
In the figure,
AB ⊥ AD. (∵ tangent at a point and radius through the point are perpendicular to each other.)
From figure,
∠ABD = ∠ABC = 50°
In △ABD,
∠ABD + ∠BDA + ∠DAB = 180°
⇒ 50° + ∠BDA + 90° = 180°
⇒ ∠BDA + 140° = 180°
⇒ ∠BDA = 180° - 140°
⇒ ∠BDA = 40°.
From figure,
∠ADC = ∠BDA = 40°.
Arc AC subtends ∠AOC at the centre and ∠ADC on point D.
∴ ∠AOC = 2∠ADC (∵ angle subtended at centre by an arc is double the angle subtended at remaining part of circle.)
∠AOC = 2 × 40° = 80°.
Hence, value of ∠AOC = 80°.
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