Two right triangles, PQR and STU, are similar. In triangle PQR, the right angle is at vertex Q, and \(\sin(\mathrm{P})...

1. TRANSLATE the problem information

• Given information:
  • Right triangle PQR with right angle at Q
  • \(\mathrm{sin(P) = 0.8}\)
  • Right triangle STU with right angle at T
  • Triangles are similar: P↔S, Q↔T, R↔U
  • Area of triangle STU = 150 square units
• Find: Length of hypotenuse SU

2. INFER the triangle type from the sine ratio

• Since \(\mathrm{sin(P) = 0.8 = \frac{4}{5}}\), we have \(\mathrm{\frac{opposite}{hypotenuse} = \frac{4}{5}}\)
• In triangle PQR: \(\mathrm{\frac{QR}{PR} = \frac{4}{5}}\)
• This suggests we might have a 3-4-5 right triangle (a common Pythagorean triple)

3. INFER the complete side ratio using Pythagorean theorem

• If \(\mathrm{QR = 4k}\) and \(\mathrm{PR = 5k}\), then: \(\mathrm{PQ^2 + (4k)^2 = (5k)^2}\)
• \(\mathrm{PQ^2 = 25k^2 - 16k^2 = 9k^2}\)
• \(\mathrm{PQ = 3k}\)
• Triangle PQR has sides in ratio 3:4:5

4. INFER the similarity relationship

• Since triangle STU is similar to PQR, it must also have sides in ratio 3:4:5
• Let \(\mathrm{ST = 3x, TU = 4x, SU = 5x}\) for some scaling factor x

5. TRANSLATE the area condition into an equation

• Area of right triangle = \(\mathrm{\frac{1}{2} \times leg_1 \times leg_2}\)
• For triangle STU: \(\mathrm{150 = \frac{1}{2} \times ST \times TU = \frac{1}{2} \times 3x \times 4x}\)
• This gives us: \(\mathrm{150 = 6x^2}\)

6. SIMPLIFY to find the scaling factor

• \(\mathrm{6x^2 = 150}\)
• \(\mathrm{x^2 = 25}\)
• \(\mathrm{x = 5}\)

7. APPLY CONSTRAINTS and calculate final answer

• The hypotenuse \(\mathrm{SU = 5x = 5(5) = 25}\)

Answer: C. 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize that \(\mathrm{sin(P) = 0.8}\) but fail to connect this to the 3-4-5 Pythagorean triple. Instead, they might try to work with decimal values (0.8, 0.6) without recognizing the underlying integer ratio structure.

This leads them to set up more complicated calculations with decimals, often making arithmetic errors that result in selecting Choice A (15) or Choice B (20) after incorrect scaling calculations.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the 3-4-5 triangle and similarity but make algebraic errors when solving \(\mathrm{6x^2 = 150}\). Common mistakes include forgetting to take the square root or making arithmetic errors in the division.

This may lead them to select Choice D (35) if they calculate incorrectly or get confused about which dimension represents the hypotenuse.

The Bottom Line:

The key insight is recognizing that \(\mathrm{sin(P) = 0.8 = \frac{4}{5}}\) immediately signals a 3-4-5 right triangle. Students who miss this connection struggle with the entire problem, while those who see it can solve efficiently using the powerful combination of similarity and area relationships.