In triangle PQR, the measure of angle Q is 90° and QS is an altitude of the triangle. The length...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR with right angle at Q
  • \(\mathrm{PQ} = 12\) (one leg)
  • \(\mathrm{QR} = 16\) (other leg)
  • QS is altitude from Q to hypotenuse PR
  • Find the ratio \(\frac{\mathrm{QR}}{\mathrm{QS}}\)

2. INFER the solution strategy

• To find \(\frac{\mathrm{QR}}{\mathrm{QS}}\), I need to determine QS first
• Since QS is the altitude to the hypotenuse, I need to know the hypotenuse length
• Strategy: Find hypotenuse → Find altitude → Calculate ratio

3. Find the hypotenuse using Pythagorean theorem

• \(\mathrm{PR}^2 = \mathrm{PQ}^2 + \mathrm{QR}^2 = 12^2 + 16^2 = 144 + 256 = 400\)
• Therefore \(\mathrm{PR} = 20\)

4. APPLY the right triangle altitude formula

• For the altitude to the hypotenuse: \(\mathrm{QS} = \frac{\mathrm{PQ} \times \mathrm{QR}}{\mathrm{PR}}\)
• \(\mathrm{QS} = \frac{12 \times 16}{20} = \frac{192}{20} = \frac{48}{5}\)

5. SIMPLIFY the final ratio calculation

• \(\frac{\mathrm{QR}}{\mathrm{QS}} = 16 \div \frac{48}{5}\)
• Convert division by fraction: \(16 \times \frac{5}{48} = \frac{80}{48}\)
• Reduce the fraction: \(\frac{80}{48} = \frac{5}{3}\)

Answer: (C) 5/3

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing the altitude formula for right triangles

Students may try to use basic trigonometry or attempt to find QS through coordinate geometry, leading to complex calculations that don't match any answer choice. Without the direct altitude formula, they often get stuck in lengthy approaches and end up guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Making arithmetic errors in the final ratio calculation

Students correctly find \(\mathrm{PR} = 20\) and \(\mathrm{QS} = \frac{48}{5}\), but then incorrectly calculate \(16 \div \frac{48}{5}\). Common mistakes include treating it as \(16 \div \frac{48}{5} = \frac{16}{48} \times 5 = \frac{5}{3} \times 5 = \frac{25}{3}\), or getting confused with the fraction division steps. This may lead them to select Choice (E) \(\frac{4}{3}\) or Choice (D) \(\frac{5}{4}\).

The Bottom Line:

This problem tests whether students know the specific altitude formula for right triangles and can execute multi-step fraction operations accurately. The geometric insight about altitude relationships is more crucial than computational complexity.