Question

Find the radius of the smaller circle if the bigger circle with centre 'O' has radius of 1cm.

Answer 1

To Find :-

The radius of the smaller circle if the bigger circle with center 'O' has radius of 1 cm.

Construction :-

A line segment OP is drawn joining the centers of both the circles.

Solution :-

Let the radius of the bigger circle be R = 1 cm, and the radius of the smaller circle be r.

(Refer to the given attachment)

OD = OC = OR = R = 1 cm   (radius of the bigger circle)

PA = PR = PB = r  (radius of the smaller circle)

Now,

OR = OP + PR

⇒ R = OP + r

⇒ OP = R - r

Hence, P is the midpoint of OR.

∠OAP and ∠PBO are 90° and adjacent sides are of same length (i.e. r), so AOBP is a square.

OP is the diagonal of AOBP.

OP = √2 × side of the square

⇒ OP = √2 × r

⇒ OP + r = OR

⇒ √2r + r = 1

⇒ r(√2 + 1) = 1

⇒ r = 1 / (√2 + 1)

Taking √2 as 1.414

⇒ r = 1 / (1.414 + 1)

⇒ r = 1 / 2.414

⇒ r = 0.414