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

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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>.  
 
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>.  
  
−
<math>\sqrt{(1-1)^2 + (-\frac{\sqrt{3}}{3} - (1 + \frac{\sqrt{3}}{3}))^2}</math>
+
<math>\sqrt{(1-1)^2 + \left(-\frac{\sqrt{3}}{3} - \left(1 + \frac{\sqrt{3}}{3}\right)\right)^2}</math>
  
−
<math>\sqrt{(-\frac{\sqrt{3}}{3} - 1 - \frac{\sqrt{3}}{3})^2}</math>
+
<math>\sqrt{\left(-\frac{\sqrt{3}}{3} - 1 - \frac{\sqrt{3}}{3}\right)^2}</math>
  
 
The length of one side is <math>1 +  \frac{2\sqrt{3}}{3}</math>
 
The length of one side is <math>1 +  \frac{2\sqrt{3}}{3}</math>
  
 
The perimeter of the triangle is <math>3 * (1 +  \frac{2\sqrt{3}}{3})</math>, so the answer is <math>\boxed{\textbf{(D) }3 + 2\sqrt{3}}</math>
 
The perimeter of the triangle is <math>3 * (1 +  \frac{2\sqrt{3}}{3})</math>, so the answer is <math>\boxed{\textbf{(D) }3 + 2\sqrt{3}}</math>

Revision as of 19:24, 4 February 2015

Problem

A line that passes through the origin intersects 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 the other given line is $\frac{\sqrt{3}}{3}$ so the third must be $-\frac{\sqrt{3}}{3}$. Since this third 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, $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.

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

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

The length of one side is $1 + \frac{2\sqrt{3}}{3}$

The perimeter of the triangle is $3 * (1 + \frac{2\sqrt{3}}{3})$, so the answer is $\boxed{\textbf{(D) }3 + 2\sqrt{3}}$

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