Difference between revisions of "2017 AMC 10B Problems/Problem 24"
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The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? | The vertices of an equilateral triangle lie on the hyperbola <math>xy=1</math>, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? | ||
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− | ==Solution 1== | + | ==Solution 1 (Law of Cosines)== |
− | + | WLOG, let the centroid of <math>\triangle ABC</math> be <math>I = (-1,-1)</math>. The centroid of an equilateral triangle is the same as the circumcenter. It follows that the circumcircle must intersect the graph exactly three times. Therefore, <math>A = (1,1)</math>, so <math>AI = BI = CI = 2\sqrt{2}</math>, so since <math>\triangle AIB</math> is isosceles and <math>\angle AIB = 120^{\circ}</math>, then by the [[Law of Cosines]], <math>AB = 2\sqrt{6}</math>. Alternatively, we can use the fact that the circumradius of an equilateral triangle is equal to <math>\frac {s}{\sqrt{3}}</math>. Therefore, the area of the triangle is <math>\frac{(2\sqrt{6})^2\sqrt{3}}4 = 6\sqrt{3}</math>, so the square of the area of the triangle is <math>\boxed{\textbf{(C) } 108}</math>. | |
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+ | Note: We could’ve also noticed that the centroid divides the median into segments of ratio <math>2:1.</math> ~peelybonehead | ||
==Solution 2== | ==Solution 2== |
Latest revision as of 16:43, 29 September 2023
Contents
Problem
The vertices of an equilateral triangle lie on the hyperbola , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
Diagram
Solution 1 (Law of Cosines)
WLOG, let the centroid of be
. The centroid of an equilateral triangle is the same as the circumcenter. It follows that the circumcircle must intersect the graph exactly three times. Therefore,
, so
, so since
is isosceles and
, then by the Law of Cosines,
. Alternatively, we can use the fact that the circumradius of an equilateral triangle is equal to
. Therefore, the area of the triangle is
, so the square of the area of the triangle is
.
Note: We could’ve also noticed that the centroid divides the median into segments of ratio ~peelybonehead
Solution 2
Without loss of generality, let the centroid of be
. Then, one of the vertices must be the other curve of the hyperbola. Without loss of generality, let
. Then, point
must be the reflection of
across the line
, so let
and
, where
. Because
is the centroid, the average of the
-coordinates of the vertices of the triangle is
. So we know that
. Multiplying by
and solving gives us
. So
and
. So
, and finding the square of the area gives us
.
Solution 3
Without loss of generality, let the centroid of be
and let point
be
. It is known that the centroid is equidistant from the three vertices of
. Because we have the coordinates of both
and
, we know that the distance from
to any vertice of
is
. Therefore,
. It follows that from
, where
and
,
using the formula for the area of a triangle with sine
. Because
and
are congruent to
, they also have an area of
. Therefore,
. Squaring that gives us the answer of
.
Solution 4
Without loss of generality, let the centroid of be
. Assuming we don't know one vertex is
we let the vertices be
Since the centroid coordinates are the average of the vertex coordinates, we have that and
We also know that the centroid is the orthocenter in an equilateral triangle, so Examining slopes, we simplify the equation to
. From the equation
we get that
. These equations are starting to resemble Vieta's:
are the roots of the equation
. This factors as
for the points
. The side length is clearly
, so the square of the area is
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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.