AMC 10 · 2019 · #19

Grade 11 geometry-2d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking
law-of-sinespythagorean-identitylaw-of-cosinesratio-proportion convert-to-algebraextreme-principleidentify-subproblems ↑ Prerequisites: pythagorean-identitylaw-of-sines
📏 Medium solution 💡 3 insights
Problem
A triangle has all three side lengths equal to whole numbers, and all three of its angles are pinned down by their cosines. Among every triangle fitting this description, find the smallest possible perimeter.

Pick an answer.

(A)
9
(B)
12
(C)
23
(D)
27
(E)
44
How to solve
Strategy Organize Information in More Ways

The problem hands over angle information (cosines) but asks for length information (a perimeter), so the data has to be re-stored in a form that talks about sides. The bridge is the Law of Sines, which compares sides to sines rather than cosines, so the first move is to rewrite each given cosine as a sine using sin² x + cos² x = 1. Once the sines are known, the ratio a : b : c falls out immediately, and the shape of the triangle is completely determined. That leaves exactly one free choice, the scale factor, so the "least perimeter" part becomes a small whole-number question: shrink the scale as far as the integer condition allows.

1STEP 1

Fix the shape, free the size

Fixed angles fix the shape only.

a = BC, b = CA, c = AB, (a, b, c) = k (a₀, b₀, c₀)
2STEP 2

Trade each cosine for a sine

Trade each cosine for a sine.

sin A = √(1 - (11/16)²) = 3√(15)/16, sin B = √(1 - (7/8)²) = √(15)/8, sin C = √(1 - (-1/4)²) = √(15)/4
3STEP 3

Read the side ratio off the sines

The sines give the side ratio.

a : b : c = 3√(15)/16 : √(15)/8 : √(15)/4 = 3 : 2 : 4
4STEP 4

Check the three angles really fit

Check the three angles really fit.

(9 + 4 - 16)/(2 · 3 · 2) = -1/4, (4 + 16 - 9)/(2 · 2 · 4) = 11/16, (9 + 16 - 4)/(2 · 3 · 4) = 7/8
5STEP 5

Shrink the scale to its limit

Reducing the ratio gives a perimeter of 9.

P = (3 + 2 + 4)k = 9k, gcd(3, 2, 4) = 1 → k_min = 1, P_min = 9
Answer
9
Every legal perimeter has the form 9k with k a positive integer, so the attainable perimeters are 9, 18, 27, 36, …. Of the five choices only 9 and 27 are on that list, and 27 comes from k = 3, a triangle three times too big to be the least. The other three choices, 12, 23, and 44, are not multiples of 9, so no triangle of this shape has them at all. The winning triangle 2, 3, 4 also passes every sanity test on its own: 2 + 3 ＞ 4, and the longest side 4 faces C, the one obtuse angle. A quick angle-sum check confirms the data is consistent, since -cos(A+B) = -(11/16 · 7/8 - 3√(15)/16 · √(15)/8) = -(77/128 - 45/128) = -1/4 = cos C.
💡Key takeaway

Three angles lock a triangle's shape but not its size, so turn the angles into a side ratio and then shrink that ratio to the smallest whole numbers it allows.

  • Fix the shape, free the size
  • Trade each cosine for a sine
  • Read the side ratio off the sines
  • Check the three angles really fit
  • Shrink the scale to its limit

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