Difference between revisions of "2015 AMC 10A Problems/Problem 17"
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==Solution 1== | ==Solution 1== | ||
− | 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 <math>\frac{\sqrt{3}}{3}</math> so the third must be <math>-\frac{\sqrt{3}}{3}</math>. Since this third 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. | + | Since the triangle is equilateral and one of the sides is a vertical line, the triangle must have a horizontal line of symmetry, and therefore the other two sides will have opposite slopes. The slope of the other given line is <math>\frac{\sqrt{3}}{3}</math> (which is must be, given 60 degree angle of the triangle, relative to vertical) so the third must be <math>-\frac{\sqrt{3}}{3}</math>. Since this third 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. |
<math>y = -\frac{\sqrt{3}}{3}</math> | <math>y = -\frac{\sqrt{3}}{3}</math> | ||
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==Solution 2== | ==Solution 2== | ||
− | + | Draw a line from the y-intercept of the equation <math>y=1+ \frac{\sqrt{3}}{3} x</math> perpendicular to the line <math>x=1</math>. There is a square of side length 1 inscribed in the equilateral triangle. The problem becomes reduced to finding the perimeter of an equilateral triangle with a square of side length 1 inscribed in it. The side length is <math>2\left(\frac{1}{\sqrt{3}}\right) + 1</math>. After multiplying the side length by 3 and rationalizing, you get <math>\boxed{\textbf{(D) }3 + 2\sqrt{3}}</math>. | |
+ | |||
+ | ==Solution 4== | ||
+ | Let the intersection point between the line <math>y = 1 + \frac{\sqrt{3}}{3}x</math> and the line that crosses the origin be <math>P</math>. | ||
+ | |||
+ | We drop an altitude from <math>P</math> onto the line <math>x = 1</math>. Since the overall triangle is an equilateral triangle, we are splitting the base (on <math>x=1</math>) in half. As the y-axis is parallel to the line <math>x=1</math>, the altitude from P will also split the y-axis from <math>y=0</math> to <math>y=1</math> in half. From this, we can get that the y-value of P is <math>\frac{1}{2}</math>. | ||
+ | |||
+ | Plugging this into the equation <math>y = 1 + \frac{\sqrt{3}}{3}x</math>, we get that <math>x = -\frac{\sqrt{3}}{2}</math>, and thus our height for the equilateral triangle is <math>1 + \frac{\sqrt{3}}{2}</math>. Using that, we can calculate the perimeter to be <math>\boxed{\textbf{(D) }3 + 2\sqrt{3}}</math>. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/-l1Kawq_hds | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== | ||
+ | Video Solution: | ||
+ | |||
+ | https://www.youtube.com/watch?v=2kvSRL8KMac | ||
+ | |||
+ | |||
{{AMC10 box|year=2015|ab=A|num-b=16|num-a=18}} | {{AMC10 box|year=2015|ab=A|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 15:48, 11 July 2024
Problem
A line that passes through the origin intersects both the line and the line
. The three lines create an equilateral triangle. What is the perimeter of the triangle?
Solution 1
Since the triangle is equilateral and one of the sides is a vertical line, the triangle must have a horizontal line of symmetry, and therefore the other two sides will have opposite slopes. The slope of the other given line is (which is must be, given 60 degree angle of the triangle, relative to vertical) so the third must be
. Since this third line passes through the origin, its equation is simply
. To find two vertices of the triangle, plug in
to both the other equations.
We now have the coordinates of two vertices, and
. The length of one side is the distance between the y-coordinates, or
.
The perimeter of the triangle is thus , so the answer is
Solution 2
Draw a line from the y-intercept of the equation perpendicular to the line
. There is a square of side length 1 inscribed in the equilateral triangle. The problem becomes reduced to finding the perimeter of an equilateral triangle with a square of side length 1 inscribed in it. The side length is
. After multiplying the side length by 3 and rationalizing, you get
.
Solution 4
Let the intersection point between the line and the line that crosses the origin be
.
We drop an altitude from onto the line
. Since the overall triangle is an equilateral triangle, we are splitting the base (on
) in half. As the y-axis is parallel to the line
, the altitude from P will also split the y-axis from
to
in half. From this, we can get that the y-value of P is
.
Plugging this into the equation , we get that
, and thus our height for the equilateral triangle is
. Using that, we can calculate the perimeter to be
.
Video Solution
~savannahsolver
See Also
Video Solution:
https://www.youtube.com/watch?v=2kvSRL8KMac
2015 AMC 10A (Problems • Answer Key • Resources) | ||
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Followed by Problem 18 | |
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