2023 AMC 10A Problems/Problem 13
Contents
Problem
Abdul and Chiang are standing feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures . What is the square of the distance (in feet) between Abdul and Bharat?
Solution 1
Let and .
By the Law of Sines, we know that . Rearranging, we get that where is a function of . We want to maximize .
We know that the maximum value of , so this yields
A quick checks verifies that indeed works.
~Technodoggo
Solution 2 (no law of sines)
Help with the diagram please?
Let us begin by circumscribing the two points A and C so that the arc it determines has measure . Then the point B will lie on the circle, which we can quickly find the radius of by using the 30-60-90 triangle formed by the radius and the midpoint of segment . We will find that . Due to the triangle inequality, is maximized when B is on the diameter passing through A, giving a length of and when squared gives .
Solution 3
It is quite clear that this is just a 30-60-90 triangle as an equilateral triangle gives an answer of , which is not on the answer choices. Its ratio is , so .
Its square is then
~not_slay
~wangzrpi
Video Solution 1 by OmegaLearn
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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All AMC 10 Problems and Solutions |
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