Q14 of 52 Page 10

In two concentric circles prove that a chord of larger circle which is tangent to smaller circle is bisected at the point of contact. (CBSE 2012)



We are given two concentric circles C1 and C2 with centre O and a chord AB of the larger circle C1 which touches the smaller circle C2 at the point P (see Fig. 10.8).


We need to prove that AP = BP.


Let us join OP. Then, AB is a tangent to C2 at P and OP is its radius. Therefore,


OP⟂AB (Since Tangent at any point of circle is perpendicular to the radius through point of contact)


Now AB is a chord of the circle C1 and OP⟂AB. Therefore, OP is the bisector of the chord AB, as the perpendicular from the centre bisects the chord,


Hence AP = BP


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