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A circle of diameter 8 cm is placed in such a manner that it touches two perpendicular lines. Then another smaller circle is placed in the gap such that it touches the lines and the circle. What is the diameter of the smaller circle?

A circle of diameter 8 cm is placed in such a manner that it touches two perpendicular lines. Then another smaller circle is placed in the gap such that it touches the lines and the circle. What is the diameter of the smaller circle? Correct Answer 8(3 - 2√2) cm

Given:

Diameter = 8 cm

Formula Used:

Radius = Diameter/2

Calculation:[ alt="F2 Ashish.K 26-05-2020 Savita D10" src="//storage.googleapis.com/tb-img/production/20/06/F2_Ashish.K_26-05-2020_Savita_D10.png" style="width: 358px; height: 359px;">

In the figure:

A and B are the centres of the circles.

Radius AP = AQ = 8/2 = 4 cm

Suppose radius of smaller circle = x cm

Now,

In square APOQ:

Diagonal, AO = 4√2 cm

In square BCOD:

Diagonal, BO = x√2 cm

Both the circles touch each other at point S:

So, AS = 4 cm and SB = x cm

Now, we can say:

AO = AS + SB + BO

⇒ 4√2 = 4 + x + x√2

⇒ x(1 + √2) = 4(√2 – 1)

⇒ x = 4(√2 – 1)/(√2 + 1)

⇒ x = 4(√2 – 1)2/(2 – 1)

⇒ x = 4(3 – 2√2)

∴ Diameter of smaller circle = 2x = 8(3 – 2√2) cm

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