# ABC is a triangle inside a circle of centre O in which AB = BC and ∠ABC = 40°. The tangents on points A and C meet at point P, then what is the value of ∠APC?

1. 100°
2. 80°
3. 120°
4. 90°

Option 1 : 100°

## Detailed Solution

Meet point O with A and C.

Theorem: The angle subtended by an arc of a circle at its centre is twice of the angle it subtends anywhere on the circle's circumference.

∠AOC = 80°

∠OAC = ∠OCA

In ΔAOC:

∠AOC + ∠OAC + ∠OCA = 180°

∠OAC = (180° – 80°)/2 = 50°

∠CAP = 90° – 50° = 40°

PA = PA (Tangents from the same point)

In ΔAPC:

∠APC + ∠PAC + ∠PCA = 180°

∠APC = 180° – 40° – 40° = 100°
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