In parallelogram PQRS, diagonal PR has length 25. The altitude from Q to diagonal PR is h_1, and the altitude...

1. TRANSLATE the problem information

• Given information:
  • Parallelogram PQRS with diagonal PR of length 25
  • Area of triangle PQR = 90
  • Area of triangle PSR = 50
  • \(\mathrm{h_1}\) = altitude from vertex Q to diagonal PR
  • \(\mathrm{h_2}\) = altitude from vertex S to diagonal PR
  • Need to find: \(\mathrm{h_1/h_2}\)

2. INFER the key geometric relationship

• The diagonal PR divides the parallelogram into exactly two triangles: PQR and PSR
• Both triangles share the same base: the diagonal PR with length 25
• Each triangle has a different altitude (\(\mathrm{h_1}\) and \(\mathrm{h_2}\)) to this shared base

3. APPLY the triangle area formula to find \(\mathrm{h_1}\)

• For triangle PQR: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• Substitute known values: \(90 = \frac{1}{2} \times 25 \times \mathrm{h_1}\)
• SIMPLIFY: \(90 = 12.5 \times \mathrm{h_1}\)
• Solve for \(\mathrm{h_1}\): \(\mathrm{h_1} = 90/12.5 = 7.2\)

4. APPLY the triangle area formula to find \(\mathrm{h_2}\)

• For triangle PSR: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• Substitute known values: \(50 = \frac{1}{2} \times 25 \times \mathrm{h_2}\)
• SIMPLIFY: \(50 = 12.5 \times \mathrm{h_2}\)
• Solve for \(\mathrm{h_2}\): \(\mathrm{h_2} = 50/12.5 = 4\)

5. SIMPLIFY to find the final ratio

• Calculate \(\mathrm{h_1/h_2} = 7.2/4 = 1.8\)
• Convert to fraction form: \(1.8 = 18/10 = 9/5\)

Answer: C. 9/5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what \(\mathrm{h_1}\) and \(\mathrm{h_2}\) represent, thinking they might be altitudes to different sides of the parallelogram rather than both being altitudes to the same diagonal PR.

This leads to confusion about how to set up the area equations, causing them to either get stuck completely or attempt to use different bases for each triangle. They may end up guessing randomly among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up both area equations but make arithmetic errors when calculating the individual altitudes or their ratio.

For example, they might calculate \(\mathrm{h_1} = 90/12.5 = 7.2\) correctly but then make an error like \(\mathrm{h_2} = 50/12.5 = 5\) instead of 4, leading to \(\mathrm{h_1/h_2} = 7.2/5 = 1.44\), which doesn't match any answer choice. This may lead them to select Choice B (4/9) by incorrectly inverting their calculated ratio.

The Bottom Line:

Success on this problem requires recognizing that both triangles formed by the diagonal share the same base, then carefully executing the arithmetic to find each altitude and their ratio.