AMC 10 · 2007 · #10

Grade 6 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates
area-trianglescoordinate-geometry convert-to-algebra ↑ Prerequisites: area-triangles
📏 Short solution 💡 2 insights
Problem
Two points B and C are fixed in a plane. Consider every point A for which triangle ABC has area exactly 1. Describe the shape formed by the collection of all such points A.

Pick an answer.

(A)
two parallel lines
(B)
a parabola
(C)
a circle
(D)
a line segment
(E)
two points

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Drawing segment BC and asking where the apex A can sit (Tool #1) turns an abstract description into a picture of a moving point. Naming the fixed base length and the altitude (Tool #4) converts the area condition into a single equation that fixes the height. Once the shape is identified, Tool #3 (Eliminate Possibilities) checks it against each answer choice, since the problem is asking us to name one of five shapes.

1STEP 1

Write the area with BC as base

Take the fixed segment BC as the base. Area is half base times height, and since BC never changes, only the height from A can vary.

[△ ABC]=1/2 · BC · h
2STEP 2

Fixing the area fixes the height

Set the area to 1 with base b=BC. Then 1/2·b·h=1 gives h=2/b, one fixed number, so every apex A sits exactly that far from line BC.

1/2 b h=1 → h=2/b (a constant)
3STEP 3

Collect every point at that fixed distance

Every point at distance 2/b from line BC lies on one of two lines parallel to BC, one per side. Both are unbounded, so the answer is (A).

S={A:dist(A, BC)=2/b}=two lines ∥ BC
Answer
two parallel lines
Test the shape with a concrete case: put B=(0,0) and C=(2,0), so b=2 and the required height is h=2/2=1. Any apex like (0,1),(5,1),(-3,1) gives base 2 and height 1, hence area 1 — these all lie on the line y=1. The points (0,-1),(4,-1) give area 1 too and lie on y=-1. Two horizontal lines y=1 and y=-1 appear, both parallel to BC on the x-axis, exactly matching two parallel lines and ruling out a bounded shape.
💡Key takeaway

Fixing the base and the area of a triangle locks the apex to one distance from the base line, and every point that far from a line makes two parallel lines.

  • Write the area with BC as base
  • Fixing the area fixes the height
  • Collect every point at that fixed distance

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