2016 AMC 8 Problem 25
Attempt Problem 25 of the 2016 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
25.
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?
Answer: B
Small hint:
Draw the radius to the point where the semicircle touches a side of the triangle.
Big hint:
Compare the area of the right half of the triangle using two different bases.
Video solution:
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Written solution:
Let be the center of the circle, which is the midpoint of
We then get that via the Pythagorean theorem.
The right half of the isosceles triangle has area The radius is perpendicular to the tangent side , so the same area is Hence
Thus, B is the correct answer.
Problem 25 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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