Question

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 ?

Answer 1

(1) It is given that line AB is tangent to the circle at A.

∴ ∠CAB = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)

Thus, the measure of ∠CAB is 90º.



(2) Distance of point C from AB = 6 cm (Radius of the circle)

(3) ∆ABC is a right triangle.

CA = 6 cm and AB = 6 cm

Using Pythagoras theorem, we have

$$\begin{gathered} {\mathrm{BC}}^2={\mathrm{AB}}^2+{\mathrm{CA}}^2 \\ \Rightarrow \mathrm{BC}=\sqrt{6^2+6^2} \\ \Rightarrow \mathrm{BC}=6\sqrt{2}\mathrm{cm} \end{gathered}$$
Thus, d(B, C) = $6\sqrt{2}$ cm

(4) In right ∆ABC, AB = CA = 6 cm

∴ ∠ACB = ∠ABC (Equal sides have equal angles opposite to them)

Also, ∠ACB + ∠ABC = 90º (Using angle sum property of triangle)

∴ 2∠ABC = 90º

⇒ ∠ABC = $\frac{90°}{2}$ = 45º

Thus, the measure of ∠ABC is 45º.