2023 AMC 10A Problems/Problem 13
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 Chaing 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 .