14. Two circles $C_{1}$ and $C_{2}$ with centers $A$ and $B$ are externally tangent at $T$. Let $B D$ be a segment tangent to $C_{1}$ at $D$ and let $T C$ be the segment tangent to both at $T$ with $C \in B D$. If $A T$ is 80 units long and $B T$ is 90 units long, what is the length of $C D$?
14. The answer is 48. $A T$ is a radius of the circle $C_{1}$; since $A D$ is also a radius, it measures 80. Furthermore, the angle $A \widehat{D} B$ is a right angle; by the Pythagorean theorem, then $B D$ measures $\sqrt{(80+90)^{2}-80^{2}}=150$. But the triangles $A B D$ and $B T C$ are similar (they are right triangles and have the angle at $B$ in common), so $B D: B T = B A: B C$ and $B C$ measures 102, therefore the length of $C D$ is 48.

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