Let A BC D be a quadrilateral with area 18 , with side A B parallel to the side C D and A B = 2 C D . Let A D be perpendicular to A B and C D . If a circle is drawn inside the quadrilateral A BC D tou

Question

Let A BC D be a quadrilateral with area 18 , with side A B parallel to the side C D and A B = 2 C D . Let A D be perpendicular to A B and C D . If a circle is drawn inside the quadrilateral A BC D touching all the sides, then its radius is

TestHub · Accepted Answer

18 = 2 1 ​ ( 3 α ) ( 2 r ) ⇒ α r = 6 Line, y = − α 2 r ​ ( x − 2 α ) is tangent to circle ( x − r ) 2 + ( y − r ) 2 2 α r ​ = r 2 = 3 r and α = 6 = 2 ​ 2 1 ​ ( x + 2 x ) × 2 r x r tan θ tan ( 9 0 ∘ − θ ) r x − r ​ ​ = 18 = 6 = r x − r ​ = r 2 x − r ​ = 2 x − r r ​ × ( 2 x − 3 r ) = 0 ​ x = 2 3 r ​ From Eqs. (i) and (ii), we get r = 2

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