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 C_{1} and C_{2} 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 C_{1} 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 C_{2}

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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