AMC 10 · 2013 · #8

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

coordinate-geometryarea-trianglessystems-of-equationsslope-intercept convert-to-algebraidentify-subproblems ↑ Prerequisites: coordinate-geometryarea-triangles

📏 Medium solution 💡 2 insights

Problem

Three lines through known points bound a triangle of given area. Find the third line's slope.

Pick an answer.

(A)

$\frac{2}{3}$

(B)

$\frac{3}{4}$

(C)

(D)

$\frac{4}{3}$

(E)

$\frac{3}{2}$

How to solve

Strategy Introduce a Variable

The only thing free to move is C, which slides along the horizontal line y = 1. Name its x-coordinate. Then the base BC is horizontal and the height from A is locked at a single number, so the area condition becomes one equation in one letter. That equation has two roots, so the last job is to decide honestly which one the positive-slope condition allows, and then check the survivor really produces area 3.

1STEP 1

Find B by substitution

Substitution finds the first crossing point.

3x - 2(1) = 1 ⟹ x = 1 ⟹ B = (1,1)

2STEP 2

Use the horizontal side as base

The horizontal side makes a clean base.

h = |1 - (-2)| = 3, [ABC] = 1/2 · BC · 3

3STEP 3

Solve for C's position

The area gives two candidate positions.

1/2 |c-1| · 3 = 3 ⟹ |c-1| = 2 ⟹ c = 3 or c = -1

4STEP 4

Rule out the vertical candidate

One would make the line vertical.

c = -1 ⟹ x_C = x_A = -1 ⟹ line x = -1, slope undefined

5STEP 5

Compute the slope and check it

The other gives slope 3/4, choice (C).

m = (1 - (-2))/(3 - (-1)) = 3/4 ＞ 0, [ABC] = 1/2 · 2 · 3 = 3

Answer

3/4

Plot the three points: A = (-1,-2), B = (1,1), C = (3,1). The top side runs along y = 1 from x = 1 to x = 3, so it has length 2, and A sits 3 units below that line, giving area 1/2 · 2 · 3 = 3 as required. The slope 3/4 is positive and is choice (B). A steepness check agrees: l₁ has slope 3/2 and climbs from A to y = 1 in a horizontal run of 2; l₃ is flatter, so it needs a longer run, and indeed it takes a run of 4 to reach y = 1, landing at x = 3 to the right of B. Every answer choice is positive, so the choices themselves give no shortcut, which is a sign the case analysis genuinely had to be done.

💡Key takeaway

When one side of a triangle lies on a horizontal line, use that side as the base: the height stops changing, and the area then tells you exactly how long the base must be.

Find B by substitution
Use the horizontal side as base
Solve for C's position
Rule out the vertical candidate
Compute the slope and check it

Find B by substitution

Use the horizontal side as base

Solve for C's position

Rule out the vertical candidate

Compute the slope and check it

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