In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD.

Let sides AB, BC, CD, and AD touches circle at P, Q, R and S respectively.



As we know that tangents drawn from an external point to a circle are equal ,


In the given image we have,


AP = AS = w (say) [Tangents from point A]


BP = BQ = x (say) [Tangents from point B]


CP = CR = y (say) [Tangents from point C]


DR = DS = z (say) [Tangents from point D]


Now,


Given,


AB = 6 cm


AP + BP = 6


w + x = 6 [1]


BC = 9 cm


BP + CP = 9


x + y = 9 [2]


CD = 8 cm


CR + DR = 8


y + z = 8 [3]


Also,


AD = AS + DS = w + z [4]


Add [1] and [3] and substracting [2] from the sum we get,


w + x + y + z - (x + y) = 6 + 8 - 9


w + z = 5 cm


From [4]


AD = 5 cm


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