Question 574556: In a 5 x 12 rectangle, a diagonal is drawn and circles are inscribed in both of the right triangles formed. what is the distance between the centers of these circles.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! 
An easy way to find this distance is to assume that A is the origin of some xy coordinate system. Therefore we may assume that A = (0,0), B = (12,0), C = (12,5), D = (0,5).
First we should find r. There are several ways to do this, but the easiest way is to use the area formula for triangle ACD:
where r is the inradius (shown above) and s is the semi-perimeter of ACD ((5+12+13)/2 = 15) Plugging in, we have
Since the two radii meet at right angles with AD and DC, the x-coordinate of the center of the upper-left circle is simply 2. The y-coordinate is 5-2, or 3, so the coordinates of this circle are (2,3).
Similarly, the coordinates of the center of the lower-right circle are (12-2, 2) or (10,2) (by symmetry). We use the distance formula to find the distance between the two centers:
^2 + (2-3)^2} = \sqrt{65}) .
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