A circle having centre c1, radius r1 = 5 cm is placed against a right angle. another smaller circle having centre c2, radius r2 is also placed touching the sides of angle and the bigger circle as shown in the figure. Find the radius r2, in cm, of the smaller circle.

Correct option is C

Given:

A circle having centre c₁, radius r₁= 5 cm is placed against a right angle. 

smaller circle centre c₂, radius r₂is also placed touching the sides of angle and the bigger circle.

Formula used:

Using Pythagoras theorem:

In ΔOCB, OC²= OB²+ BC²

Solution:  

In ΔOCB, OC² = OB² + BC²

OC²= 25 + 25

OC = 5√2

In ΔOAP, OP²= OA²+ AP²

OP²= r₂²+ r₂²

OP = r₂√2

Now OC = OP + PC

5√2 = r₂√2 + (PQ + CQ)

5√2 = r₂√2 + r₂+ 5

5(√2 - 1) = r₂(√2 + 1)

​$$\begin{aligned}& r_2=\frac{5(\sqrt{2}-1)} {(\sqrt{2}+1)} \\& r_2=\frac{5(2+1-2 \sqrt{2})}{(2-1)} \quad-\cdots \text { (rationalising the terms) } \\& r_2=5(3-2 \sqrt{2})\end{aligned}$$ 

Thus, the correct answer is (c).