A circle is inscribed in a quadrilateral ABCD in such a manner that the circle is tangent to all the four sides of the quadrilateral. It is given that Angle BAD = 90⁰= Angle CBA.

Find the radius of the circle given that: BC = 21 units and AD = 28 units.

What would be the radius of the inscribed circle if BC = 36 units and AD = 45 units?

Let a = |AD|, b = |BC|, and r = radius of circle.

For a tangential quadrilateral we have

  |AB| + |CD| = |AD| + |BC|

             or 

  |CD| = |AD| + |BC| - |AB| = a + b - 2r     (1)

Applying the Pathagorean theorem we have

  |CD|^2 = |AB|^2 + (|AD| - |BC|)^2

         = (2r)^2 + (a - b)^2                (2)

Combining (1) and (2) gives

        a*b  
  r = -------
       a + b

Case  I: a = 28 and b = 21 gives r = 12.

Case II: a = 45 and b = 36 gives r = 20.

Posted by Bractals on 2006-12-31 12:23:31