Question

In the given figure, three circles with centres A, B, C respectively touch each other externally. If AB = 5 cm, BC = 7 cm and CA = 6 cm then the radius of the circle with centre A is

(a) 1.5 cm 

(b) 2 cm 

(c) 2.5 cm 

(d) 3 cm

Answer 1

Correct answer is (b) 2 cm

Given, AB = 5 cm, BC = 7 cm and CA = 6 cm.

Let, AR = AP = x cm.

BQ = BP = y cm

CR = CQ = z cm

(Since the length of tangents drawn from an external point arc equal)

Then, AB = 5 cm

\(\Rightarrow\) AP + PB = 5 cm

\(\Rightarrow\) x + y = 5   ......(i)

Similarly, y + z = 7   ....(ii)

and z + x = 6   .........(iii)

Adding (i), (ii) and (iii), we get:

(x + y) + (y + z) + (z + x) = 18

\(\Rightarrow\) 2(x + y + z) = 18

\(\Rightarrow\) (x + y + z) = 9    ........(iv)

Now, (iv) – (ii):

\(\Rightarrow\) x = 2

∴ The radius of the circle with center A is 2 cm.