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Subj:.....Circle Inscribed In Triangle

          From: MathNexus.wwu.edu

          on 5/24/2008 (S648)

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Source: http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=167

The perimeter of a right triangle is 324 cm and its hypotenuse
is 135 cm.  Find the radius of the circle that can be inscribed
in the triangle.
 

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Hint: The solution path can be either complicated or straight-
forward....which approach are you trying? 

For the straight-forward approach, look at the picture (i.e. circle
incribed in a right triangle), then stand back and look at it....
what geometrical "laws" do you see in action?

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Finger pointing down
from darrell94590 on 1/2/2006
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Drawing from Ripleys-Believe It Or Not
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              THE SOLUTION
 

Let X and Y be the two legs of the right triangle,

then  X  +  Y  +  135  =  324

and   X² +  Y² =  135²

Solving the two equations and two unknowns using
the quadratic equation yields the two sides of
81 cm. and 108 cm.

Remember that the circle breaking the triangle into
three equal pairs tangents to a circle from it's
vertices.
 
Let a, b, and c  be the sides of the
   three pairs of equal tangents,

then  a  +  b  =  81

and   b  +  c  = 135

also  a  +  c  = 108

Solving yields a  =  27 cm.
               b  =  54 cm.
         and   c  =  81 cm.
Noticing that the two "a"s and the "r" are three sides
of a square, r = 27 cm.

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