Question

in a quadrilateral abcd,it is given that ab=ad+13, bc=cd=20, bd=24. if r is the radius of the circlr inscriable in the quadrilateral,then what is the integer closest to it

Answer 1

In the given figure, it can be observed that AB touches the circle at point P; BC touches the circle at point Q; CD touches the circle at point R;  DA touches the circle at point S. 
Then, we can say from a theorem that "Length of tangents drawn from an external point to the circle are same ", 
Therefore, 
DR = DS (Tangents on the circle from point D) ….. (1) 
CR = CQ (Tangents on the circle from point C) …... (2)
BP = BQ (Tangents on the circle from point B) ….... (3)
AP = AS (Tangents on the circle from point A) ….... (4)
Adding all these equations, we obtain
DR + CR + BP + AP = DS + CQ + BQ + AS
(DR + CR) + (BP + AP) = (DS + AS) + (CQ + BQ)
 CD + AB = AD + BC 
Hence, Proved.
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