Difference between revisions of "2023 AMC 10A Problems/Problem 13"
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<i>~Education, the Study of Everything </i> | <i>~Education, the Study of Everything </i> | ||
Latest revision as of 14:21, 14 August 2024
Contents
- 1 Problem
- 2 Video Solution by MegaMath
- 3 Solution 1
- 4 Solution 2 (no law of sines)
- 5 Solution 3 (Guessing)
- 6 Solution 4
- 7 Solution 5
- 8 Solution 6 (Logic)
- 9 Solution 7
- 10 Video Solution 🚀 Under 2 min 🚀
- 11 Video Solution by Power Solve
- 12 Video Solution by SpreadTheMathLove
- 13 Video Solution 1 by OmegaLearn
- 14 Video Solution by CosineMethod
- 15 Video Solution
- 16 Video Solution by Math-X
- 17 See Also
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?
Video Solution by MegaMath
https://www.youtube.com/watch?v=ZsiqPRWCEkQ&t=3s
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 check verifies that indeed works.
~Technodoggo ~(minor grammar edits by vadava_lx)
Solution 2 (no law of sines)
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 (Guessing)
Guess that the optimal configuration is 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
Solution 4
We use ,
,
to refer to Abdul, Bharat and Chiang, respectively.
We draw a circle that passes through
and
and has the central angle
.
Thus,
is on this circle.
Thus, the longest distance between
and
is the diameter of this circle.
Following from the law of sines, the square of this diameter is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 5
We can represent Abdul, Bharat and Chiang as ,
, and
, respectively.
Since we have
and (from other solutions)
, this is a
triangle.
By the side ratios of a
triangle, we can infer that
.
Squaring AB we get
.
~ESAOPS
Solution 6 (Logic)
As in the previous solution, refer to Abdul, Bharat and Chiang as ,
, and
, respectively- we also have
. Note that we actually can't change the lengths, and thus the positions, of
and
, because that would change the value of
(if we extended either of these lengths, then we could simply draw
such that
is perpendicular to
, so
is unchanged). We can change the position of
to alter the values of
and
, but throughout all of these changes,
remains unvaried.
Therefore, we can let
.
(What is the justification for all of these assumptions??)
It follows that is
-
-
, and
.
is then
and the square of
is
.
-Benedict T (countmath1)
Solution 7
(why?)
We look at the answers and decide: the square of
is
.
-vvsss
Video Solution 🚀 Under 2 min 🚀
~Education, the Study of Everything
Video Solution by Power Solve
https://www.youtube.com/watch?v=jkfsBYzBJbQ
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=nmVZxartc-o
Video Solution 1 by OmegaLearn
Video Solution by CosineMethod
https://www.youtube.com/watch?v=BJKHsHQyoTg
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by Math-X
https://youtu.be/N2lyYRMuZuk?si=_Y5mdCFhG-XD7SaG&t=631
~Math-X
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.