In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is a tangent at A. Answer the following questions.

(1) What is the measure of ∠CAB ? Why?

(2) What is the distance of point C from line AB? Why?

(3) d(A,B) = 6 cm, find d(B,C).

(4) What is the measure of ∠ABC ? Why?

(1) ere CA is the radius of the circle and A is the point of contact of the tangent AB.

⇒ ∠CAB = 90° Using tangent-radius theorem which states that a tangent at any point of a circle is perpendicular to the radius at the point of contact.

(2) CA is the radius of the circle which is perpendicular to the tangent AB.

So, the perpendicular distance of line AB from C = CA = 6 cm

(3) In triangle ABC right-angled at A,

Given AB = 6 cm and CA = 6 cm

BC² = AB² + CA² {Using Pythagoras theorem}

⇒ BC² = 6² + 6²

⇒ BC² = 36 + 36

⇒ BC = √72

⇒ BC = 6√2 cm

(4) In triangle ABC right-angled at A,

AB = CA = 6 cm

⇒∠ABC = ∠ACB {Angles opposite to equal sides are equal}

⇒∠ABC + ∠ACB + ∠ BAC = 180° {Angle sum property of the triangle}

⇒ 2∠ABC = 90° {∵ ∠ BAC = 90°}

⇒ ∠ABC = 45°