Problem 6 · 2017 AMC 8
Easy
Geometry & Measurement
ratiosubstitution
If the degree measures of the angles of a triangle are in the ratio 3 : 3 : 4, what is the degree measure of the largest angle of the triangle?
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Answer: D — 72°.
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Hint 1 of 2
A ratio doesn't give angles directly — it gives shares. Picture the 180° of a triangle cut into 3 + 3 + 4 = 10 equal pieces; the question is how big one piece is.
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Hint 2 of 2
The 'parts' trick: add the ratio numbers to get the total parts, divide the known whole by that to size one part, then scale up to the part you want.
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Approach: parts of a whole
- The angles of any triangle add to 180°. The ratio splits that 180° into 3 + 3 + 4 = 10 equal parts, so one part = 180 ÷ 10 = 18°. This sizing step is the key — everything else is multiplication.
- The largest angle holds 4 parts: 4 × 18 = 72°.
- Sanity check: 54 + 54 + 72 = 180. You'll reuse this 'sum-of-parts' idea for any "divide a total in ratio a:b:c" problem — money, lengths, mixtures, all the same move.
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