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
\[{BC}^2 = {AB}^2 + {CA}^2 \]
\[ \Rightarrow BC = \sqrt{6^2 + 6^2} \]
\[ \Rightarrow BC = 6\sqrt{2} cm\]
Thus, d(B, C) = \[6\sqrt{2}\]
(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 = \[\frac{90^\circ}{2}\]
Thus, the measure of ∠ABC is 45º.