Difference between revisions of "2023 AIME I Problems/Problem 8"
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
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Revision as of 21:59, 8 February 2023
Problem
Rhombus has . There is a point on the incircle of the rhombus such that the distances from P to the lines and are and , respectively. Find the perimeter of .
Solution 1
Denote by the center of . We drop an altitude from to that meets at point . We drop altitudes from to and that meet and at and , respectively. We denote . We denote the side length of as .
Because the distances from to and are 16 and 9, respectively, and , the distance between each pair of two parallel sides of is . Thus, and .
We have
Thus, .
In , we have . Thus,
Taking the imaginary part of this equation and plugging and into this equation, we get
We have
Because is on the incircle of , . Plugging this into (1), we get the following equation
By solving this equation, we get and . Therefore, .
Therefore, the perimeter of is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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