In the figure, PA and PC are tangents to the circle ABC. If ∠P = 48°, then ∠ABC = ?
Explanation:
Draw two lines joining A and C with center O.
Now in quadrilateral AOCP, sum of all four angles should be 360°.
Also, ∠OAP = ∠OCP = 90° (radius is line joining center and point of tangency is perpendicular to the tangent)
∴ 90° + ∠AOC + 90° + 48° = 360°
⇒ ∠AOC = 132°
Angle subtended by chord AC in the major segment is half of the angle it subtends at the center.
∴ ∠ADC = ½ × ∠AOC = 66°
Now, angles subtended by a chord in different segments are supplementary.
∴ ∠ADC + ∠ABC = 180°
⇒ ∠ABC = 180° - 66° = 114°.
Hence, 114.
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