Difference between revisions of "2020 AMC 8 Problems/Problem 18"
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+ | ==Solution 3 (coordinate bashing)== | ||
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+ | Let the midpoint of segment <math>FE</math> be the origin. Evidently, point <math>D</math> is at <math>(-8, 0)</math> and <math>A</math> is at <math>(8, 0)</math>. Since points <math>C</math> and <math>B</math> share x-coordinates with <math>D</math> and <math>A</math>, respectively, we can just find the y-coordinate of <math>B</math> (which is just the width of the rectangle) and multiply this by <math>DA</math>, or <math>16</math>. Since the radius of the semicircle is <math>\frac{9+16+9}{2}</math>, or <math>17</math>, the equation of the circle that our semicircle is a part of is <math>x^2+y^2=289</math>. Since we know that the x-coordinate of <math>B</math> is <math>8</math>, we can plug this into our equation to obtain that <math>y=\pm15</math>. Since <math>y>0</math>, as the diagram suggests, we know that the y-coordinate of <math>B</math> is <math>15</math>. Therefore, our answer is <math>16\cdot 15</math>, or <math>\boxed{\textbf{(A) }240}</math>. | ||
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+ | 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. | ||
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+ | - StarryNight7210 | ||
==See also== | ==See also== |
Revision as of 14:01, 18 November 2020
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 | |
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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