Circle Inscribed in TriangleDate: 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/ |
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