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 An inscribed circle problem (Posted on 2006-12-31)
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)

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 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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