Difference between revisions of "2023 AMC 10A Problems/Problem 13"
(Solution 2) |
m (format) |
||
Line 17: | Line 17: | ||
~Technodoggo | ~Technodoggo | ||
− | ==Solution 2== | + | ==Solution 2 (no law of sines)== |
− | |||
Help with the diagram please? | Help with the diagram please? | ||
− | Let us begin by circumscribing the two points A and C so that the arc it determines has measure <math>120</math>. 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 \overline{ | + | Let us begin by circumscribing the two points A and C so that the arc it determines has measure <math>120</math>. 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 <math>\overline{AC}</math>. We will find that <math>r=16*\sqrt3</math>. Due to the triangle inequality, <math>\overline{AB}</math> is maximized when B is on the diameter passing through A, giving a length of <math>32*\sqrt3</math> and when squared gives <math>\boxed{\text{(C) }3072}</math>. |
Revision as of 21:03, 9 November 2023
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
.