AMC 10 · 2006 · #10

Grade 7 geometry-2d
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
polygon-inequalitylinear-equations-one-varoptimization bound-inequality-then-enumerate ↑ Prerequisites: polygon-inequality
📏 Medium solution 💡 2 insights
Problem
A triangle has whole-number side lengths. One side is three times the length of a second side, and the third side is 15. Find the largest perimeter such a triangle can have.

Pick an answer.

(A)
43
(B)
44
(C)
45
(D)
46
(E)
47

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The question asks for the greatest possible perimeter, and since the perimeter is 4x+15, that means pushing x as large as the rules allow — a textbook use of Tool #14 (Extreme Principle): the answer lives at the top boundary. To find where that boundary is, Tool #4 (Introduce a Variable) names the short related side x, writes the sides as x, 3x, 15, and turns the triangle condition into inequalities. Tool #3 (Eliminate Possibilities) then rejects any x that is too big to close into a real triangle.

1STEP 1

Name the sides with one letter

Call the shorter related side x. The sides are then x, 3x, and 15, so the perimeter is 4x+15.

sides=x, 3x, 15, perimeter=4x+15
2STEP 2

Write the triangle condition

Each side must be shorter than the other two combined: x+3x ＞ 15 and x+15 ＞ 3x. The third check, 3x+15 ＞ x, is automatic.

x+3x ＞ 15, x+15 ＞ 3x
3STEP 3

Solve the two inequalities for x

Simplifying gives x ＞ 3.75 and x ＜ 7.5, so the whole-number values left are 4, 5, 6, 7.

4x ＞ 15→ x ＞ 3.75; 15 ＞ 2x→ x ＜ 7.5; x∈{4,5,6,7}
4STEP 4

Take the largest x and add up the sides

The biggest legal value x=7 gives sides 7, 21, 15, and 7+15=22 ＞ 21 holds — perimeter 43, choice (A).

x=7: 7+21+15=43 (7+15=22 ＞ 21) → (A)
Answer
43
The winning triangle has sides 7, 21, 15, all whole numbers, with 21=3 × 7 and a third side of 15, so it fits every condition. The two short sides just clear the long one, 7+15=22 ＞ 21, meaning the triangle is barely valid — exactly what you expect at the maximum. All the larger answer choices 44,45,46,47 would need x ＞ 7, which fails the triangle inequality, so 43 is the only reachable one.
💡Key takeaway

When a problem asks for the biggest possible answer, find the boundary the rules allow and push right up to it — here the triangle inequality caps the short side at 7, giving perimeter 43.

  • Name the sides with one letter
  • Write the triangle condition
  • Solve the two inequalities for x
  • Take the largest x and add up the sides

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