Circle Inscribed in Triangle

Date: 04/04/97 at 10:29:27
From: Anonymous
Subject: Radius of Circle Inscribed in Right Triangle

A circle is inscribed in a right triangle with sides 3, 4 and 5. What 
is the radius of the circle? I know the radius forms a 90 degree angle 
with the tangent line but other than that I haven't a clue.

Thanks

Date: 04/04/97 at 11:53:50
From: Doctor Wilkinson
Subject: Re: Radius of Circle Inscribed in Right Triangle

Draw a picture of the triangle ABC with the right angle at C and 
with BC measuring 4, AC measuring 3, and AB measuring 5.  Let O be the 
center of the inscribed circle and draw the 3 radii perpendicular to 
the three sides of the triangle. Let the ends of these radii be D on 
BC, E on AC, and F on AB.

Finally, draw in the line segments OA and OB.

   

Now we're going to stare at this picture for a while.  

Let's let r be the unknown radius.  Now you have a little square OECD 
in the picture, right?  You know it's a square because of the right 
angles at E, C, and D and the equal radii OE and OD.  So you have CD 
measuring r and therefore DB measuring 4 - r.

Similarly, you have EC measuring r and therefore EA measuring 3 - r.

That's about all the information we can get out of the little square.  
What else have we got?

We've got a couple of triangles ODB and OFB, and they're congruent!  
(I'll leave it to you to prove this part).  So FB is congruent 
to DB and measures 4 - r.  

Similarly, FA measures 3 - r.

But FA and FB add up to AB which measures 5.  So now you have an 
equation for r.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/