Question

Abdul and Chiang are standing $48$ 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 $60^\circ.$ What is the square of the distance (in feet) between Abdul and Bharat?

$1728$

$2601$

$3072$

$4608$

$6912$

Solution:

Let $A$ be Abdul, $C$ be Chiang with $AC = 48,$ and $B$ be Bharat with $\angle B = 60^\circ.$ Every point seeing $AC$ at $60^\circ$ lies on one circular arc, so all valid $B$ sit on a circle where chord $AC$ subtends $60^\circ.$ The law of sines gives its diameter, $\frac{AC}{\sin 60^\circ} = \frac{48}{\sqrt3/2} = 32\sqrt3.$ Now $AB$ is a chord, and a chord is longest when it's a diameter. So $AB = 32\sqrt3$ and $AB^2 = 1024 \cdot 3 = 3072.$ Thus, C is the correct answer.

2023 AMC 10A Problem 13

Solution:

Problem 13 in Other Years