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In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6 cm, BC = 7 cm and CD = 4 cm. Find 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 = 7 cm
BP + CP = 7
x + y = 7 ….[2]
CD = 4 cm
CR + DR = 4
y + z = 4 ….[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 + 4 - 7
w + z = 3 cm ; From [4]
AD = 3 cm