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 sides, one three times another, and a third side of 15. Find the largest possible perimeter.

Pick an answer.

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

The ratio writes all three sides with one letter.

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

Write the triangle condition

Two inequalities are genuinely needed, one from each side.

x+3x ＞ 15, x+15 ＞ 3x
3STEP 3

Solve the two inequalities for x

Solving them and keeping whole values leaves four options.

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 largest option gives 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

More like this

Same archetype — closest grade level first.