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Math Help - Mutually Tangent Triangles

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by dunafrothint View Post
    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.
    Attached Thumbnails Attached Thumbnails Mutually Tangent Triangles-dreikreistangent.png  
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