The measures of the three angles of a triangle are in the ratio 17 : 13 : 15. Find the positive difference between the greatest and the smallest of these three angles.

Correct — C. The three angles of a triangle add to 180°, so the ratio terms share that 180°.

Sum of terms: 17 + 13 + 15 = 45
One part: 180 ÷ 45 = 4°
Greatest angle: 17 × 4 = 68°
Smallest angle: 13 × 4 = 52°

Difference: 68 − 52 = 16° — option (c).

Faster route: the extremes are 17 − 13 = 4 parts apart, and 4 parts × 4° = 16°.

Why the others are wrong

• (a)
  24° — 24° is 6 parts of 4°. It would be right only if a part were 6°, which needs the terms to sum to 30 — they sum to 45.
• (b)
  20° — 20° needs the extremes 5 parts apart. Compute them instead: 68° and 52°, a gap of 16°.
• (d)
  12° — 12° is 3 parts. The ratio terms 17, 13 and 15 are 2, 2 and 4 parts apart from one another, never 3.

A ratio fixes a triangle's angles completely, because their sum is known to be 180°.

Add the terms: 17 + 13 + 15 = 45. One part is therefore 180 ÷ 45 = 4°.

Every angle, and every difference between two angles, is then a whole number of parts times 4°. The greatest and smallest terms are 17 and 13, so their gap is 4 parts, or 16° — the individual angles never have to be written down.

Key facts

• The angles of a triangle sum to 180°, so a ratio a : b : c makes one part 180/(a + b + c).
• Here one part is 4°, giving angles of 68°, 52° and 60°.
• A difference of two angles equals the difference of their ratio terms times the value of one part.
• All three angles are under 90°, so this triangle is acute.

Common traps

• Assuming the printed order is descending — the smallest term here is 13, in the middle
• Dividing 180 by 3 instead of by 45, the sum of the terms
• Answering 4, the difference of the ratio terms, without multiplying by the 4° a part is worth

No directly related past PYQ was found.