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A semicircle is inscribed in an isosceles triangle with base and height so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

2016 AMC (American Mathematical Contests) 8 problem 25

Number Theory

Easy

Do you really need a hint? Try it first!

Hint figure :

Verify \( AD = 8 \) and recall the concepts of Similar Triangles. Then proceed .

Verify \( \triangle AED \sim \triangle ACD \) . And proceed to find the radius of the semicircle i.e. DE . [NOTE : As DE is the radius and AC be the tangent so \( DE \perp AC \) .]

We have \( \triangle AED \sim \triangle ACD \) . This similarity means that we can create a proportion: \( \frac {AD}{AC} = \frac {DE}{CD} \) . We plug in \( AD = 8 , AC = 17 \ and \ CD = 15 \) . So \( DE = \frac {8}{17} \times 15 = \frac {120}{17} \) .

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