AMC 10 · 2017 · #12

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometrylinear-equations-two-varinterval-arithmetic caseworkspatial-visualization ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
Look at the three quantities 3, x+2, and y-4. A point (x,y) belongs to the set S when two of these three quantities are equal to each other and the remaining one is less than or equal to that shared value. Describe the shape that all such points (x,y) form in the plane.

Pick an answer.

(A)
a single point
(B)
two intersecting lines
(C)
three lines whose pairwise intersections are three distinct points
(D)
a triangle
(E)
three rays with a common endpoint

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The phrase "two of the three quantities are equal" hides three separate possibilities — which two are the equal pair. Tool #7 (Identify Subproblems) splits the question into those three clean cases. In each case, setting the two equal quantities equal gives one straight-line equation, while the "third is no greater than the common value" clause adds one inequality that cuts that line down to a half-line. Tool #4 (Introduce a Variable) lets us treat 3, x+2, y-4 as named quantities so we can write each equation and inequality, and Tool #1 (Draw a Diagram) lets us see where the three pieces sit — the punchline is whether they share an endpoint.

1STEP 1

Split into three equal-pair cases

Name A=3, B=x+2, C=y-4. "Two equal" gives exactly three cases: A=B, A=C, B=C, each leaving the third quantity ≤ the common value.

A=3, B=x+2, C=y-4; cases: A=B, A=C, B=C
2STEP 2

Case A=B: the vertical ray

Set 3=x+2, so x=1; then y-4 ≤ 3 gives y ≤ 7. That keeps the vertical line x=1 only below, a ray from (1,7) going down.

x=1, y-4 ≤ 3 → y ≤ 7 → ray from (1,7) downward
3STEP 3

Case A=C: the horizontal ray

Set 3=y-4, so y=7; then x+2 ≤ 3 gives x ≤ 1. That keeps the horizontal line y=7 only leftward, a ray from (1,7) going left.

y=7, x+2 ≤ 3 → x ≤ 1 → ray from (1,7) leftward
4STEP 4

Case B=C: the slanted ray

Set x+2=y-4, i.e. y=x+6; then 3 ≤ x+2 gives x ≥ 1. Keep that slanted line only for x ≥ 1, a ray from (1,7) going up-right.

x+2=y-4 → y=x+6, 3 ≤ x+2 → x ≥ 1 → ray from (1,7)
5STEP 5

Combine the three rays

All three pieces are rays sharing the endpoint (1,7): down, left, and up-right. Three rays from one common endpoint is description (E).

three rays, all with endpoint (1,7) → (E)
Answer
three rays with a common endpoint
Every case produced a straight-line equation trimmed by one inequality, so each piece must be a ray, never a full line or a bounded segment — this already rules out choices (B), (C), and (D). Checking the shared corner: setting x=1,y=7 gives the three quantities 3, 3, 3, all equal, so (1,7) lies on all three rays and is their common endpoint. The three rays leave that corner in three different directions (down, left, up-right) without overlapping, so S is not a single point either. Three rays with one common endpoint matches (E).
💡Key takeaway

When "two of three are equal," try each pair in turn — here each pair makes a line that the leftover condition trims to a ray, and all three rays hinge on the same corner (1,7).

  • Split into three equal-pair cases
  • Case A=B: the vertical ray
  • Case A=C: the horizontal ray
  • Case B=C: the slanted ray
  • Combine the three rays

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