Question

In fig. 1, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°, find ∠PCA.

Answer 1

∠PCA = 60°

Step-by-step explanation:

Given:

Figure provided in attachment below.

PQ is a tangent at a point C to a circle with centre O.

AB is a diameter and  ∠CAB = 30°

To Find:

∠PCA = ?

Calculating:

Consider Δ ACO,

AO = OC (Radii of the same circle)

∴ Δ ACO is an isosceles triangle.

It is given that ∠CAB = 30°

∠CAO = ∠ACO = 30°

(Angles opposite to equal sides in an isosceles triangle are equal)

∠PCO = 90° (Tangent ⊥ Radius)

∠PCA = ∠PCO - ∠CAO

∴ ∠PCA = 90° - 30° = 60°

Therefore, the value of ∠PCA = 60°.