Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.

Let C1 and C2 are two concentric circles with center O and radius of outer circle is 5 cm.

Given : AC is a chord with length 8 cm that is tangent to inner circle .

To find : Radius of inner circle i.e. OD

AC is a tangent for C1 at D so,

OD AC [Tangent at a point on the circle is perpendicular to the radius through point of contact ]

So, OAD is a right-angled triangle at D .

Also it implies that OD is perpendicular to the chord AC in C2

So we have,

AD = DC [perpendicular from the center to the chord bisects the chord]

AC = AD + DC

8 = AD + AD
AD = 4 cm

In OAD By Pythagoras Theorem

(OA)2 = (OD)2 + (AD)2

(5)2 = (OD)2+ (4)2

25 = (OD)2 + 16

(OD)2= 9

OD = 3 cm