2017 AMC 8 Problems/Problem 22
In the right triangle , , , and angle is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
We can reflect triangle over line This forms the triangle and a circle out of the semicircle. Let us call the center of the circle . We can see that Circle is the incircle of We can use the formula for finding the radius of the incircle to solve this problem: Area of a triangle . The area of is The semiperimeter is Simplifying Our answer is therefore a
Asymptote diagram by Mathandski
We immediately see that , and we label the center of the semicircle and the point where the circle is tangent to the triangle . Drawing radius with length such that is perpendicular to , we immediately see that because of congruence, so and . By similar triangles and , we see that .
Let the center of the semicircle be . Let the point of tangency between line and the semicircle be . Angle is common to triangles and . By tangent properties, angle must be degrees. Since both triangles and are right and share an angle, is similar to . The hypotenuse of is , where is the radius of the circle. (See for yourself) The short leg of is . Because ~ , we have and solving gives
Let the tangency point on be . Note By Power of a Point, Solving for gives
https://youtu.be/3VjySNobXLI - Happytwin
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