AMC 10 · 2017 · #9

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Vertices then Area
coordinate-geometrylinear-equations-two-varinterval-arithmetic caseworkidentify-subproblems ↑ Prerequisites: coordinate-geometrylinear-equations-two-var
📏 Medium solution 💡 3 insights
Problem
Two of three quantities must tie and the third must not exceed them. Describe the resulting shape.

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 point
How to solve
Strategy Identify Subproblems

The condition names a pair without saying which pair, so Tool #7 (Identify Subproblems) is the natural entry: there are only three possible pairs, and each one turns the sentence into plain algebra — one equation plus one inequality. Solving three easy subproblems beats trying to handle the vague wording all at once. The equation in each case describes a line and the inequality keeps only part of it, so Tool #1 (Draw a Diagram) is what turns three algebraic answers into one picture. Finally the question is multiple choice about a shape, so Tool #3 (Eliminate Possibilities) is used at the end to test the finished picture against each description rather than trusting a first impression.

1STEP 1

Split into the three possible ties

There are exactly three possible ties.

A = 3, B = x+2, C = y-4; the three cases are A = B ≥ C, A = C ≥ B, and B = C ≥ A
2STEP 2

Case one: the tie is 3 and x+2

The first tie gives a vertical piece.

3 = x+2 → x = 1; y-4 ≤ 3 → y ≤ 7; piece = {(1, y) : y ≤ 7}
3STEP 3

Case two: the tie is 3 and y-4

The second gives a horizontal piece.

3 = y-4 → y = 7; x+2 ≤ 3 → x ≤ 1; piece = {(x, 7) : x ≤ 1}
4STEP 4

Case three: the tie is x+2 and y-4

The third gives a slanted piece.

x+2 = y-4 → y = x+6; 3 ≤ x+2 → x ≥ 1 (equivalently y ≥ 7); piece = {(x, x+6) : x ≥ 1}
5STEP 5

All three pieces stop at the same point

All three stop at the same point.

3 = x+2 = y-4 → x = 1 and y = 7 → (x,y) = (1,7)
6STEP 6

Compare the picture to the choices

So the shape is three rays, choice (E).

S = {(1,y) : y ≤ 7} ∪ {(x,7) : x ≤ 1} ∪ {(x,x+6) : x ≥ 1} → (E)
Answer
three rays with a common point
Spot-check one point on each piece. At (1,0) the quantities are 3, 3, -4: the first two tie and -4 ≤ 3, so it is in S, on the downward ray. At (-5,7) they are 3, -3, 3: the outer two tie and -3 ≤ 3, so it is in S, on the leftward ray. At (4,10) they are 3, 6, 6: the last two tie at 6 and 3 ≤ 6, so it is in S, on the slanted ray. The cut-offs point the right way too: at (2,7) the quantities are 3, 4, 3 — a tie at 3, but the leftover 4 is greater than 3, so the horizontal piece really does stop at x=1 rather than continuing right. Likewise (1,8) gives 3, 3, 4 and fails, so the vertical piece really does stop at y=7. One caution worth naming: reversing the inequality would swing all three rays to the opposite directions yet still leave three rays from one point, so the letter (E) alone would not have caught a direction error — these checks are what confirm the directions.
💡Key takeaway

"Two are equal and the third is no bigger" just means the biggest value is tied, so try each pair in turn — every tie gives a line, and the leftover quantity trims that line down to a ray.

  • Split into the three possible ties
  • Case one: the tie is 3 and x+2
  • Case two: the tie is 3 and y-4
  • Case three: the tie is x+2 and y-4
  • All three pieces stop at the same point
  • Compare the picture to the choices

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