# Radius of a circle that intersects two points on a right triangle.

1. May 18, 2011

### dgoldman86

I'm trying to figure out the radius of a circle that intersects two points on a right triangle. One side of the triangle is tangent to the circle and the other intersects it. I have attached an image that helps further explain what I'm talking about. Knowing what I have listed in the image is there anyway to determine the radius of that circle?

Thanks.

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• ###### Circle.JPG
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2. May 19, 2011

### henry_m

Try using coordinate geometry. Let the point where the triangle is tangent to the circle be the origin, and let that side of the circle run parallel to the x-axis. What's the equation of the circle? What are the coordinates of the point where the triangle touches the circle?

3. May 22, 2011

### jorgealnino

How about this

Call the vertex of your right triangle $(c_1,r,c2)$ with $c_1$ tangent to the circle and $r$ the vertex at the right angle. Let $o$ be the center of your circle.
Now construct the square $(o,c_1,r,s)$.
Observe that $c_2$ is on the square side $(r,q)$ and use pythagoras.

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