AMC 10 · 2005 · #24

Grade 11 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

coordinate-geometryequilateral-triangletangent-addition-formulavieta-formulas convert-to-algebravieta-formulas ↑ Prerequisites: coordinate-geometryslope-interceptequilateral-triangle

📏 Long solution 💡 4 insights

Problem

Three points on a parabola form an equilateral triangle, and one side has slope two. The three horizontal coordinates add to a fraction in lowest terms. Find the sum of its numerator and denominator.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

A point of y = x² is pinned down by its x-coordinate alone, so name the three coordinates a, b, c. On this parabola the slope of a chord collapses to the sum of the two x-coordinates, which means the quantity being asked for is exactly half the sum of the three side slopes. That reframing lets the equilateral condition be written as one quadratic equation in an unknown slope, and the sum of that quadratic's two roots is all that is needed. The vertices themselves never have to be found.

1STEP 1

Name the three vertices

Naming the three points reduces the triangle to three numbers.

A = (a, a²), B = (b, b²), C = (c, c²)

2STEP 2

A chord's slope is a sum

A chord's slope is simply the sum of two coordinates.

slope of AB = (b² - a²)/(b - a) = (b-a)(b+a)/(b-a) = a + b

3STEP 3

The target is half the slope sum

So the target is exactly half the total of the three side slopes.

(a+b) + (b+c) + (c+a) = 2(a+b+c) ⟹ a+b+c = (m_AB + m_BC + m_CA)/2

4STEP 4

Equilateral means two 60-degree turns

Equilateral means the directions sit sixty degrees apart.

tanθ = 2, other two directions: θ + 60° and θ - 60°

5STEP 5

Turn the 60 degrees into an equation

The tangent formula turns that into one quadratic.

|(m - 2)/(1 + 2m)| = tan 60° = √(3) ⟹ (m-2)² = 3(1+2m)²

6STEP 6

The two unknown slopes are its two roots

The two unknown slopes are exactly its two roots.

m² - 4m + 4 = 12m² + 12m + 3 ⟹ 11m² + 16m - 1 = 0, Δ = 256 + 44 = 300 ＞ 0

7STEP 7

Vieta gives the sum for free

Vieta adds them without computing either, giving 3/11.

m₁ + m₂ = -16/11, Σ slopes = 2 - 16/11 = 6/11, a+b+c = 1/2·6/11 = 3/11

8STEP 8

Reduce and add

It is already reduced, so the answer is 14, choice (A).

m/n = 3/11, gcd(3,11) = 1, m + n = 3 + 11 = 14

Answer

Build the triangle and measure it. The quadratic's roots are (-8 ± 5√3)/11, so with a + b = 2 the third coordinate is c = 3/11 - 2 = -19/11, and subtracting each root from 3/11 gives a = 1 - 5√3/11 ≈ 0.2127 and b = 1 + 5√3/11 ≈ 1.7873. The three side lengths of the triangle with vertices (a,a²), (b,b²), (c,c²) all come out to ≈ 3.5209, and the three slopes are 2, ≈ -1.5146, ≈ 0.0600. So a triangle meeting every condition genuinely exists, it really is equilateral, and its x-coordinates really sum to 0.2727… = 3/11. The sum being a small positive number is also sensible: two vertices sit near x = 1 and the third sits near x = -19/11, so heavy cancellation is expected.

💡Key takeaway

On y = x² the slope of a line joining two points is just their x-coordinates added, so a question about points turns into a question about slopes.

Name the three vertices
A chord's slope is a sum
The target is half the slope sum
Equilateral means two 60-degree turns
Turn the 60 degrees into an equation
The two unknown slopes are its two roots
Vieta gives the sum for free
Reduce and add