Difference between revisions of "2017 AMC 10B Problems/Problem 24"
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− | ==Problem | + | ==Problem== |
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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</asy> | </asy> | ||
− | ==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>. | |
+ | |||
+ | |||
+ | 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== | ||
− | + | WLOG, let the centroid of <math>\triangle ABC</math> be <math>G = (-1,-1)</math>. Then, one of the vertices must be the other curve of the hyperbola. WLOG, let <math>A = (1,1)</math>. Then, point <math>B</math> must be the reflection of <math>C</math> across the line <math>y=x</math>, so let <math>B = \left(a,\frac{1}{a}\right)</math> and <math>C=\left(\frac{1}{a},a\right)</math>, where <math>a <-1</math>. Because <math>G</math> is the centroid, the average of the <math>x</math>-coordinates of the vertices of the triangle is <math>-1</math>. So we know that <math>a + 1/a+ 1 = -3</math>. Multiplying by <math>a</math> and solving gives us <math>a=-2-\sqrt{3}</math>. So <math>B=(-2-\sqrt{3},-2+\sqrt{3})</math> and <math>C=(-2+\sqrt{3},-2-\sqrt{3})</math>. So <math>BC=2\sqrt{6}</math>, and finding the square of the area gives us <math>\boxed{\textbf{(C) } 108}</math>. | |
+ | <math>\newline</math> | ||
+ | Alternatively, from <math>a + \frac{1}{a} = -4</math>, we obtain <math>a^2 - 2 + \frac{1}{a^2} = (-4)^2 - 4 \implies \left|a - \frac{1}{a} \right| = \sqrt{12}</math>. Since <math>BC</math> lies on the line <math>x + y = -4</math>, which forms an isosceles right triangle with the coordinate axes, <math>BC = \sqrt{12} \sqrt{2} = \sqrt{24}</math>. Hence the squared area is <math>\left(\frac{\sqrt{3}}{4} s^2 \right)^2 = \frac{3}{16} \cdot 24^2 = \boxed{108}</math>. | ||
==Solution 3== | ==Solution 3== | ||
− | Without loss of generality, let the centroid of <math>\triangle ABC</math> be <math>G = (1, 1)</math> and let point <math>A</math> be <math>(-1, -1)</math>. It is known that the centroid is equidistant from the three vertices of <math>\triangle ABC</math>. Because we have the coordinates of both <math>A</math> and <math>G</math>, we know that the distance from <math>G</math> to any | + | Without loss of generality, let the centroid of <math>\triangle ABC</math> be <math>G = (1, 1)</math> and let point <math>A</math> be <math>(-1, -1)</math>. It is known that the centroid is equidistant from the three vertices of <math>\triangle ABC</math>. Because we have the coordinates of both <math>A</math> and <math>G</math>, we know that the distance from <math>G</math> to any vertex of <math>\triangle ABC</math> is <math>\sqrt{(1-(-1))^2+(1-(-1))^2} = 2\sqrt{2}</math>. Therefore, <math>AG=BG=CG=2\sqrt{2}</math>. It follows that from <math>\triangle ABG</math>, where <math>AG=BG=2\sqrt{2}</math> and <math>\angle AGB = \dfrac{360^{\circ}}{3} = 120^{\circ}</math>, <math>[\triangle ABG]= \dfrac{(2\sqrt{2})^2 \cdot \sin(120)}{2} = 4 \cdot \dfrac{\sqrt{3}}{2} = 2\sqrt{3}</math> using the formula for the area of a triangle with sine <math>\left([\triangle ABC]= \dfrac{1}{2} AB \cdot BC \sin(\angle ABC)\right)</math>. Because <math>\triangle ACG</math> and <math>\triangle BCG</math> are congruent to <math>\triangle ABG</math>, they also have an area of <math>2\sqrt{3}</math>. Therefore, <math>[\triangle ABC] = 3(2\sqrt{3}) = 6\sqrt{3}</math>. Squaring that gives us the answer of <math>\boxed{\textbf{(C) }108}</math>. |
+ | |||
+ | ==Solution 4== | ||
+ | Without loss of generality, let the centroid of <math>\triangle ABC</math> be <math>G = (1, 1)</math>. Assuming we don't know one vertex is <math>(-1, -1)</math> we let the vertices be <math>A\left(x_1, \frac{1}{x_1}\right), B\left(x_2, \frac{1}{x_2}\right), C\left(x_3, \frac{1}{x_3}\right).</math> | ||
+ | |||
+ | Since the centroid coordinates are the average of the vertex coordinates, we have that <math>\frac{x_1+x_2+x_3}{3}=1</math> and <math>\frac{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}}{3}=1.</math> | ||
+ | |||
+ | We also know that the centroid is the orthocenter in an equilateral triangle, so <math>CG \perp AB.</math> Examining slopes, we simplify the equation to <math>x_1x_2x_3 = -1</math>. From the equation <math>\frac{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}}{3}=1,</math> we get that <math>x_1x_2+x_1x_3+x_2x_3 = -3</math>. These equations are starting to resemble Vieta's: | ||
+ | |||
+ | <math>x_1+x_2+x_3=3</math> | ||
+ | |||
+ | <math>x_1x_2+x_1x_3+x_2x_3 = -3</math> | ||
+ | |||
+ | <math>x_1x_2x_3=-1</math> | ||
+ | |||
+ | <math>x_1,x_2,x_3</math> are the roots of the equation <math>x^3 - 3x^2 - 3x + 1 = 0</math>. This factors as <math>(x+1)(x^2-4x+1)=0 \implies x = -1, 2 \pm \sqrt3,</math> for the points <math>(-1, -1), (2+\sqrt3, 2-\sqrt3), (2-\sqrt3, 2+\sqrt3)</math>. The side length is clearly <math>\sqrt{24}</math>, so the square of the area is <math>\boxed{108}.</math> | ||
+ | <math>\sim\textbf{Leonard\_my\_dude}\sim</math> | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=B|num-b=23|num-a=25}} | {{AMC10 box|year=2017|ab=B|num-b=23|num-a=25}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 04:06, 15 October 2024
Contents
[hide]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
WLOG, let the centroid of be . Then, one of the vertices must be the other curve of the hyperbola. WLOG, 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 . Alternatively, from , we obtain . Since lies on the line , which forms an isosceles right triangle with the coordinate axes, . Hence the squared area is .
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 vertex 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 | |
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