Competition · AMC preparation · step 4 of 4

AMC 8 · 2017 · #6

Grade 8 geometry-2drate-ratio
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up
angle-sum-triangleratio-proportion identify-subproblemsratio-proportion ↑ Prerequisites: ratio-proportionmulti-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem
A triangle has three interior angles whose measures are in the ratio 3:3:4. Find the degree measure of the largest of those three angles.

Pick an answer.

(A)
18
(B)
36
(C)
60
(D)
72
(E)
90

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

Tool #7 (Identify Subproblems) splits the question into two clean pieces: (a) figure out how many degrees one "ratio unit" is worth, then (b) use that unit to size the largest angle. The ratio 3:3:4 contains 3+3+4=10 equal units total, and those 10 units must share the 180° of a triangle — so one unit is just 180° ÷ 10. Once we know one unit, the largest angle (the 4-unit part) is immediate. Tool #3 (Eliminate Possibilities) is the back-up check: since this is multiple choice, every candidate answer should be at most 180° and should be expressible as 4 copies of an integer share, which quickly rules out most options.

1STEP 1

Add the ratio parts

Add the ratio parts: 3 + 3 + 4 = 10 equal shares split the 180°.

3 + 3 + 4 = 10 parts
2STEP 2

Find one share of 180

Any triangle's three angles sum to 180°, so one share is 180° ÷ 10 = 18°.

180° ÷ 10 = 18° per share
3STEP 3

Multiply one share by 4

The largest angle is the 4-share part, so 4 × 18° = 72°.

4 × 18° = 72° → (D)
4STEP 4

Check the three angles sum

Check: 54° + 54° + 72° = 180°, which rules out every other choice.

54° + 54° + 72° = 180° ✓
Answer
72
The three angles work out to 54°, 54°, 72° — all positive, all less than 180°, and summing to exactly 180°. The triangle is isosceles (two equal small angles) with one slightly larger angle, which matches the 3:3:4 ratio (the largest is just a bit bigger than the other two, not dramatically so). The largest angle 72° is less than 90°, so this is an acute triangle — consistent with the modest 3:3:4 ratio. Answer (D) 72° is the right size.
💡Key takeaway

This AMC 8 problem only needs the Grade 8 fact you already know — the three angles of a triangle always add up to 180°!

  • Add the ratio parts
  • Find one share of 180
  • Multiply one share by 4
  • Check the three angles sum

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