AMC 10 · 2015 · #17

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometryslope-interceptequilateral-trianglethirty-sixty-ninety-triangle symmetry-argument ↑ Prerequisites: coordinate-geometryequilateral-triangle
📏 Medium solution 💡 3 insights
Problem
Three lines fence off a triangle: the vertical line x = 1, the slanted line y = 1 + (33\frac{\sqrt{3}}{3})x, and a third line through the origin. The triangle they make is equilateral. Find its perimeter.

Pick an answer.

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

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Sketching the three lines shows one side is vertical, which forces a horizontal mirror line for the equilateral triangle. That symmetry pins down the unknown third line's slope without any heavy algebra. After that, the job splits into small pieces: find the two corners that sit on x = 1, measure that vertical side, then triple it because all sides of an equilateral triangle match.

1STEP 1

Read the slope as an angle

A 30° slope against the vertical side means the two lines meet at a 60-degree angle — one corner of the triangle.

slope = 33\frac{\sqrt{3}}{3} = 13\frac{1}{\sqrt{3}} = tan 30°
2STEP 2

Use symmetry for the third line

The vertical side gives the triangle a horizontal mirror symmetry, flipping the slope to −33\frac{\sqrt{3}}{3} for the third line.

y = -33\frac{\sqrt{3}}{3} x
3STEP 3

Find the two corners on x = 1

Plugging x = 1 into both slanted lines locates the triangle's other two corners.

(1, 1+33\frac{\sqrt{3}}{3}) and (1, -33\frac{\sqrt{3}}{3})
4STEP 4

Measure the vertical side

Since both corners share x = 1, the side's length is just the gap between their y-values.

s = (1+33\frac{\sqrt{3}}{3})-(-33\frac{\sqrt{3}}{3}) = 1 + 233\frac{2\sqrt{3}}{3}
5STEP 5

Triple it for the perimeter

Equal sides mean tripling one side gives the perimeter: 3 + 2√3, matching choice (D).

P = 3(1 + 233\frac{2\sqrt{3}}{3}) = 3 + 2√3
Answer
3 + 2√(3)
Numerically 3 + 2√3 is about 6.46, so each side is roughly 2.15. The vertical side runs from y ≈ −0.58 up to y ≈ 1.58, a gap of about 2.15, which agrees. Checking the apex where the two slanted lines meet gives (−32\frac{\sqrt{3}}{2}, 12\frac{1}{2}), and its distance to either corner is also about 2.15, confirming the triangle really is equilateral.
💡Key takeaway

A vertical side makes the triangle a mirror image, so the third line just flips the slope's sign; find the corners, measure, triple.

  • Read the slope as an angle
  • Use symmetry for the third line
  • Find the two corners on x = 1
  • Measure the vertical side
  • Triple it for the perimeter

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