Length of tangent line segment to 2 circles

Question

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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?

Answer 1

Since $\triangle ACD$ is a right triangle with $CD=6$, we have $$AB=AC-BC=\sqrt{6^2-4^2}-2=2\sqrt 5-2.$$

Answer 2

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