2020 AMC 8 Problems/Problem 18
Rectangle is inscribed in a semicircle with diameter as shown in the figure. Let and let What is the area of
Solution
First, realize is not a square. It can easily be seen that the diameter of the semicircle is , so the radius is . Express the area of Rectangle as , where . Notice that by the Pythagorean theorem . Then, the area of Rectangle is equal to . ~icematrix
Solution 2
We have , as it is a radius, and since it is half of . This means that . So
~yofro
Solution 3 (coordinate bashing)
Let the midpoint of segment be the origin. Evidently, point is at and is at . Since points and share x-coordinates with and , respectively, we can just find the y-coordinate of (which is just the width of the rectangle) and multiply this by , or . Since the radius of the semicircle is , or , the equation of the circle that our semicircle is a part of is . Since we know that the x-coordinate of is , we can plug this into our equation to obtain that . Since , as the diagram suggests, we know that the y-coordinate of is . Therefore, our answer is , or .
NOTE: The synthetic solution is definitely faster and more elegant. However, this is the solution that you should use if you can't see any other easier solution.
- StarryNight7210
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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