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
[asy] draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("",(8,0), 1.25*S); dot("",(8,15), 1.25*N); dot("",(-8,15), 1.25*N); dot("",(-8,0), 1.25*S); dot("",(17,0), 1.25*S); dot("",(-17,0), 1.25*S); label("",(0,0),N); label("",(12.5,0),N); label("",(-12.5,0),N); dot("", (0,0), 1.25*N);[/asy]
We have , as it is a radius, and since it is half of . This means that . So
~yofro
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.