Q. 54.7( 21 Votes )

If the sides of a

Answer :

Given:  the sides of a quadrilateral touch a circle

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

Proof:

From the theoram which states that the lengths of the two tangents drawn from an external point to a circle are equal

From points A the tangents drawn are AP and AS,

AP = AS .... (1)

From points B the tangents drawn are BP and BQ,

BP = BQ  ..... (2)

From points D the tangents drawn are DR and DS,

DR = DS ....(3)

From points C the tangents drawn are CR and CQ,

CR = CQ  ..... (4)

Add 1,2,3 and 4 to get

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

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

AB+ DC = AD + BC

Hence proved

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