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Geometry Difficulty 3.5 AMC 10/12 Find the answer Italy

Two circles $C_{1}$ and $C_{2}$ with centers $A$ and $B$ are externally tangent at $T$ . Let $BD$ be a segment tangent to $C_{1}$ at $D$ , and let $TC$ be the segment tangent to both at $T$ with $C \in BD$ . If $AT$ has length 80 and $BT$ has length 90, what is the length of $CD$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The answer is 48. $AT$ is a radius of the circle $C_{1}$ ; since $AD$ is also one, it too measures 80. Moreover the angle $A \widehat{D} B$ is right; by the Pythagorean theorem then $BD$ measures $\sqrt{(80+90)^{2}-80^{2}}=150$ . But the triangles $ABD$ and $BTC$ are similar (they are right triangles and share the angle at $B$ ), so $BD : BT = BA : BC$ and $BC$ measures 102, so that the length of $CD$ is 48.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.