Q. 43.6( 10 Votes )

If the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite sides is equal to the sum of the other pair.

Given: sides of a quadrilateral touch a circle.

To prove: sum of a pair of opposite sides is equal to the sum of the other pair

Theorem Used: The length of two tangents drawn from an externa point are equal

Explanation:

Let ABCD is a quadrilateral which touches the circle at points P,Q,R and S.

Since length of a tangent drawn from external points to a circle are equal.

DR = DS ----(i)
CR = CQ-----(ii)
AP = AS-----(iii)
BP = BQ-----(iv)

Adding four equation we get,

DR+CR) +AP+BP=DS+CQ+AS+BQ

DC+AB = DA+CB

Hence Proved.

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