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If AB is a chord of a circle with center O, AOC is a diameter and AT is the tangent at A as shown in figure. Prove that ∠BAT = ∠ACB.
Given:-
A circle with center O and AC as a diameter and AB and BC as two chords also AT is a tangent at point A.
To prove:-
- ∠BAT = ∠ACB
Proof:-
∠ABC = 90° [Angle in a semicircle is a right angle]
In △ABC by angle sum property of triangle
∠ABC + ∠ BAC + ∠ACB = 180 °
∠ACB + 90° = 180° - ∠BAC
∠ACB = 90 - ∠BAC [1]
Now,
OA ⏊ AT [Tangent at a point on the circle is perpendicular to the radius through point of contact]
∠OAT = ∠CAT = 90°
∠BAC + ∠BAT = 90°
∠BAT = 90° - ∠BAC [2]
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From [1] and [2]
∠BAT = ∠ACB [Proved]
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