# In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm then find the radius of the smaller circle.

Let us consider circles C1 and C2 with common center as O. Let AB be a tangent to circle C1 at point P and chord in circle C2. Join OB

In OPB

OP AB

[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]

OPB is a right - angled triangle at P,

By Pythagoras Theorem,

[i.e. (Hypotenuse)2 = (Base)2 + (Perpendicular)2]

(OB)2 = (OP)2 + (PB)2

Now, 2PB = AB

[As we have proved above that OP AB and Perpendicular drawn from center to a chord bisects the chord]

2PB = 8 cm

PB = 4 cm

(OB)2 = (5)2 + (4)2

[As OP = 5 cm, radius of inner circle]

(OB)2 = 41

OB = √41 cm

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