# Thread: Mutually Tangent Triangles

1. ## 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!

2. 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:

$\displaystyle \overline{AB} = 8$

$\displaystyle \overline{AC} = 3+x$

$\displaystyle \overline{BC} = 5+x$

Use the cosine rule:

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

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

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