2017 AMC 8 Problems/Problem 22
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?
Solution
We can reflect triangle on 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 is The semiperimeter is Simplifying Our answer is therefore
Similar Triangles Solution
We immediately see that , and we label the center of the semicircle . Drawing radius with length such that is tangent to , we immediately see that because of HL congruence, so and . By similar triangles and , we see that .
Similar Triangles Solution 2.
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) One leg of is . Because ~ , we have and solving gives
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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