Mathematics
In the given figure, O is the center of the circle, PA is tangent and PBC is secant. If ∠ABC = 60°; ∠P is :
30°
60°
120°
90°
Circles
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Answer
In △ABC,
∠BAC = 90° (Angle in semi-circle is a right angle)
⇒ ∠ABC + ∠BAC + ∠ACB = 180° (By angle sum property of triangle)
⇒ 60° + 90° + ∠ACB = 180°
⇒ 150° + ∠ACB = 180°
⇒ ∠ACB = 180° - 150° = 30°.
We know that,
The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.
⇒ ∠BAP = ∠ACB = 30°.
From figure,
⇒ ∠PBA + ∠ABC = 180° [Linear pairs]
⇒ ∠PBA + 60° = 180°
⇒ ∠PBA = 180° - 60° = 120°.
In △PBA,
⇒ ∠PBA + ∠BAP + ∠APB = 180° (By angle sum property of triangle)
⇒ 120° + 30° + ∠APB = 180°
⇒ 150° + ∠APB = 180°
⇒ ∠APB = 180° - 150° = 30°.
∴ ∠P = 30°.
Hence, Option 1 is the correct option.
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