+448 Two circles, centered at A and B, are externally tangent to each other, and tangent to a line $\ell.$ A third circle, centered at $C,$ is externally tangent to the first two circles, and the line $\ell.$ If the radii of circle $A$ and circle $B$ are $9$ and $9,$ respectively, then what is the radius of circle $C?$
+130477 See the following
Let circle A have a radius of 9 and let the small circle have a radius , r
Triangle ACD is right with
AC =the hypotenuse = 9 + r
AD is a leg = 9
DC is another leg = 9 -r
So we have this relationship
9^2 + ( 9 - r)^2 = (9 + r)^2
81 + 81 -18r + r^2 = 81 + 18r + r^2 simplify
81 = 36r
r = 81/36 = 9/4 = the radius of the small circle
