2022 AMC 12A Problems/Problem 13
Contents
Problem
Let be the region in the complex plane consisting of all complex numbers that can be written as the sum of complex numbers and , where lies on the segment with endpoints and , and has magnitude at most . What integer is closest to the area of ?
Solution
If is a complex number and , then the magnitude (length) of is . Therefore, has a magnitude of 5. If has a magnitude of at most one, that means for each point on the segment given by , the bounds of the region could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of . Therefore, the total area is .
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Video Solution 1 (Quick and Simple)
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Video Solution 1 (Simple and Fun!!!)
https://youtu.be/7yAh4MtJ8a8?si=5AxafzIVqF71CtGL&t=2712
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See Also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 AMC 12 Problems and Solutions |
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