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: The 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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