# Mutually Tangent Triangles

• May 5th 2009, 09:56 PM
dunafrothint
Mutually Tangent Triangles
Three circles are mutually tangent, but none inside the other. If the radii of circle A is 3cm and circle B is 5cm. Then find out the radius of the third circle, given that (angle)BAC is 60(degrees).

Also, if someone could mention a decent program for drawing graphs and such, it would be greatly appreciated.

Thanks!
• May 6th 2009, 04:07 AM
earboth
Quote:

Originally Posted by dunafrothint
Three circles are mutually tangent, but none inside the other. If the radii of circle A is 3cm and circle B is 5cm. Then find out the radius of the third circle, given that (angle)BAC is 60(degrees).

Also, if someone could mention a decent program for drawing graphs and such, it would be greatly appreciated.

Thanks!

Let x denote the length of the 3rd radius.
A, B, C are the vertices of a triangle with the side lengths:

$\overline{AB} = 8$

$\overline{AC} = 3+x$

$\overline{BC} = 5+x$

Use the cosine rule:

$(\overline{AB})^2+(\overline{AC})^2-2\cdot \overline{AB} \cdot \overline{AC} \cdot \cos(60^\circ) = (\overline{BC})^2$

$(8)^2+(3+x)^2-2\cdot 8 \cdot (3+x) \cdot \dfrac12 = (5+x)^2$

Solve for x: I've got x= 2.