2020 AMC 8 Problems/Problem 18
Contents
- 1 Problem
- 2 Solution 1 (Pythagorean Theorem)
- 3 Solution 2 (Coordinate Geometry)
- 4 Solution 3
- 5 Solution 4 (Power of a Point Theorem)
- 6 Solution 5 (Vertical Theorem)
- 7 Video Solution (🚀 Quick 🚀)
- 8 Video Solution by North America Math Contest Go Go Go
- 9 Video Solution by WhyMath
- 10 Video Solution
- 11 Video Solution by Interstigation
- 12 Video Solution by OmegaLearn
- 13 Video Solution by SpreadTheMathLove
Problem
Rectangle is inscribed in a semicircle with diameter as shown in the figure. Let and let What is the area of
Solution 1 (Pythagorean Theorem)
Let be the center of the semicircle. The diameter of the semicircle is , so . By symmetry, is in fact the midpoint of , so . By the Pythagorean theorem in right-angled triangle (or ), we have that (or ) is . Accordingly, the area of is .
Solution 2 (Coordinate Geometry)
Let the midpoint of segment be the origin. Evidently, point and . Since points and share -coordinates with and respectively, it suffices to find the -coordinate of (which will be the height of the rectangle) and multiply this by (which we know is ). The radius of the semicircle is , so the whole circle has equation ; as already stated, has the same -coordinate as , i.e. , so substituting this into the equation shows that . Since at , the y-coordinate of is . Therefore, the answer is .
(Note that the synthetic solution (Solution 1) is definitely faster and more elegant. However, this is the solution that you should use if you can't see any other easier strategy.)
Solution 3
We can use a result from the Art of Problem Solving Introduction to Algebra book Sidenote: for a semicircle with diameter , such that the part is on one side and the part is on the other side, the height from the end of the side (or the start of the side) is . To use this formula, we scale the figure down by ; this will give the height a length of . Now, scaling back up by , the height is . The answer is then . -SweetMango77
Solution 4 (Power of a Point Theorem)
Draw the other half of the circle as follows: By the Power of a Point Theorem, . By symmetry, . We see that and . Substituting in these values, , giving and . The area of the rectangle is therefore .
Solution 5 (Vertical Theorem)
According to the Pythagorean Theorem and the Vertical Theorem, we can find out that . Therefore, the answer is ;
Video Solution (🚀 Quick 🚀)
~Education, the Study of Everything
Video Solution by North America Math Contest Go Go Go
https://www.youtube.com/watch?v=5Qo4pG3Uk_U
~North America Math Contest Go Go Go
Video Solution by WhyMath
~savannahsolver
Video Solution
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=852
~Interstigation
Video Solution by OmegaLearn
https://youtu.be/7SwJdAEOeAg?t=23
~ pi_is_3.14
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=S_CnKCuOA_w
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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