Triangle PQR is a right triangle with angle R a right angle. If sin P = 5/13 and the length...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR with right angle at R
  • \(\sin P = \frac{5}{13}\)
  • \(\mathrm{PR} = 40\) units
  • Need to find perimeter
• What this tells us: We have a right triangle where we know one trigonometric ratio and one side length.

2. INFER the triangle pattern

• Key insight: \(\sin P = \frac{5}{13}\) matches the 5-12-13 Pythagorean triple pattern
• Since angle R is the right angle:
  • QR is opposite to angle P
  • PR is adjacent to angle P
  • PQ is the hypotenuse
• This means the sides are in ratio \(5:12:13\)

3. Set up the scaling relationship

• If sides are in ratio \(5:12:13\), then:
  • \(\mathrm{QR} = 5k\) (opposite to P)
  • \(\mathrm{PR} = 12k\) (adjacent to P)
  • \(\mathrm{PQ} = 13k\) (hypotenuse)

4. SIMPLIFY to find the scaling factor

• Since \(\mathrm{PR} = 40\) and \(\mathrm{PR} = 12k\):

\(12k = 40\)

\(k = \frac{40}{12} = \frac{10}{3}\)

5. Calculate all side lengths

• \(\mathrm{QR} = 5k = 5\left(\frac{10}{3}\right) = \frac{50}{3}\)
• \(\mathrm{PR} = 40\) (given)
• \(\mathrm{PQ} = 13k = 13\left(\frac{10}{3}\right) = \frac{130}{3}\)

6. SIMPLIFY to find the perimeter

• Perimeter = \(\mathrm{QR} + \mathrm{PR} + \mathrm{PQ}\)
• Perimeter = \(\frac{50}{3} + 40 + \frac{130}{3}\)
• Perimeter = \(\frac{50 + 130}{3} + 40 = \frac{180}{3} + 40 = 60 + 40 = 100\)

Answer: D. 100

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\sin P = \frac{5}{13}\) indicates a 5-12-13 right triangle pattern. Instead, they try to use the Pythagorean theorem or other trigonometric relationships without recognizing the efficient pattern-based approach.

Without this key insight, students get bogged down in complex calculations involving decimals (like finding \(\cos P = \sqrt{1 - \sin^2 P} = \sqrt{\frac{144}{169}} = \frac{12}{13}\)) or struggle to set up the correct relationships between the sides. This leads to confusion and often causes them to abandon systematic solution and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which sides are opposite, adjacent, and hypotenuse relative to angle P. They might incorrectly assume that PR is the hypotenuse (since it's given) or mix up which side is opposite to angle P.

This incorrect identification leads to wrong ratios and wrong calculations. For example, if they think PR is the hypotenuse, they might set up \(13k = 40\), leading to completely different side lengths and ultimately selecting Choice A (84) or Choice B (90).

The Bottom Line:

This problem rewards students who recognize special right triangle patterns and can correctly interpret angle-side relationships in right triangles. The key is seeing the 5-12-13 pattern immediately rather than getting lost in decimal calculations.