Difference between revisions of "2015 AMC 10A Problems/Problem 17"

(Created page with "==Problem== A line that passes through the origin in tersects both the line <math> x = 1</math> and the line <math>y=1+ \frac{\sqrt{3}}{3} x</math>. The three lines create an eq...")
 
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==Solution==
 
==Solution==
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Since the triangle is equilateral and one of the sides is a vertical line, the other two sides will have opposite slopes. The slope of one of the other lines is <math>\frac{\sqrt{3}}{3}</math> so the other must be <math>-\frac{\sqrt{3}}{3}</math>. Since this other line passes through the origin, its equation is simply <math>y = -\frac{\sqrt{3}}{3}x</math>. To find two vertices of the triangle, plug in <math>x=1</math> to both the other equations.
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<math>y = -\frac{\sqrt{3}}{3}</math>
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<math>y = 1 + \frac{\sqrt{3}}{3}</math>
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We now have the coordinates of two vertices. <math>(1, -\frac{\sqrt{3}}{3})</math> and <math>(1, 1 + \frac{\sqrt{3}}{3})</math>. Apply the distance formula, <math>\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}</math>.
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<math>\sqrt{(1-1)^2 + (-\frac{\sqrt{3}}{3} - (1 + \frac{\sqrt{3}}{3}))^2}</math>
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<math>\sqrt{(-\frac{\sqrt{3}}{3} - 1 - \frac{\sqrt{3}}{3}))^2}</math>

Revision as of 17:24, 4 February 2015

Problem

A line that passes through the origin in tersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?

$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)} \ 6 \qquad\textbf{(D)} \ 3 + 2\sqrt{3} \qquad\textbf{(E)} \ 6 + \frac{\sqrt{3}}{3}$


Solution

Since the triangle is equilateral and one of the sides is a vertical line, the other two sides will have opposite slopes. The slope of one of the other lines is $\frac{\sqrt{3}}{3}$ so the other must be $-\frac{\sqrt{3}}{3}$. Since this other line passes through the origin, its equation is simply $y = -\frac{\sqrt{3}}{3}x$. To find two vertices of the triangle, plug in $x=1$ to both the other equations.

$y = -\frac{\sqrt{3}}{3}$

$y = 1 + \frac{\sqrt{3}}{3}$

We now have the coordinates of two vertices. $(1, -\frac{\sqrt{3}}{3})$ and $(1, 1 + \frac{\sqrt{3}}{3})$. Apply the distance formula, $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.

$\sqrt{(1-1)^2 + (-\frac{\sqrt{3}}{3} - (1 + \frac{\sqrt{3}}{3}))^2}$

$\sqrt{(-\frac{\sqrt{3}}{3} - 1 - \frac{\sqrt{3}}{3}))^2}$

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