Q. 14.8( 21 Votes )

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.

Answer :


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


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