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Published byLisbeth Dawsey Modified about 1 year ago

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THEOREM -14 CONVERSE OF Alternate segment Theorem Statement:- “If a line is drawn through an end point of a chord of a circle so that the angle formed with the chord is equal to the angle substended by the chord in the alternate segment, then the line is a tangent to the circle”. C B A x1 XY y1 o

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Given:- circle with centre “O” XAY is a line meeting chord AB at A. and C is the point in the other segment such that BAY = ACB. C B A x1 XY y1 o

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R.T.P:- XAY is tangent to the circle Construction:- NIL C B A x1 XY y1 o

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Proof:- suppose XAY is not tangent to the circle. Let X‘AY’ is tangent to the circle Let X‘AY’ is tangent to the circle ACB = BAY (1) C B A x1 XY y1 o

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But ACB = BAY (given) (2) From (1) & (2) BAY = BAY which is possible iff XAY coincides with XAY XAY is tangent to the circle at A. C B A x1 XY y1 o

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Conclusion:- “If a line is drawn through and end point of a chord of a circle so that the angle formed with the chord is equal to the angle substended by the chord in the alternate segment, then the line is a tangent to the circle”. C B A x1 XY y1 o

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