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An inscribed circle problem (Posted on 2006-12-31) Difficulty: 2 of 5
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?

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 1 of 2

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
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