2017 AMC 8 Problems/Problem 22
Contents
Problem
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 1
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 a formula for finding the radius of the incircle. The area of a triangle . The area of is The semiperimeter is Simplifying Our answer is therefore
Asymptote diagram by Mathandski
Solution 2
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 .
Solution 3
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
Solution 4
Let the tangency point on be . Note By Power of a Point, Solving for gives
Solution 5
Let us label the center of the semicircle and the point where the circle is tangent to the triangle . The area of = the areas of + , which means . So it gives us .----LarryFlora
Video Solution
https://youtu.be/3VjySNobXLI - Happytwin
https://youtu.be/KtmLUlCpj-I - savannahsolver
https://youtu.be/FDgcLW4frg8?t=3837 - pi_is_3.14
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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