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https://drive.google.com/file/d/0B-4lJHUDH1P5UEZ4QzNYcTNYQWs/edit?usp=sharing

The image of the problem can be accessed in the above website.

Two semicircles are tangent to each other. The semicircle with center D has a radius of 4, and the semicircle with center C has a radius of 2. Segment AC is tangent to the larger semicircle and intersects the smaller semicircle at B. What is the length of segment AB?

enter image description here

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  • $\begingroup$ post the image here, please $\endgroup$ – bubba Sep 21 '14 at 14:36
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Since $\triangle ACD$ is a right triangle with $CD=6$, we have $$AB=AC-BC=\sqrt{6^2-4^2}-2=2\sqrt 5-2.$$

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You can also use the Tangent-Secant theorem to get $(AB+2)^2=2\dot(2+8)$. Solving this yields $AB=\sqrt{10}-2$.

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