A right triangle has a hypotenuse of length 13 centimeters. The length of one of the legs is 7 centimeters...

1. TRANSLATE the problem information

• Given information:
  • Hypotenuse = 13 cm
  • One leg is 7 cm shorter than the other leg
  • x = length of longer leg (what we're solving for)

• What this tells us: The shorter leg = \(\mathrm{(x - 7)\,cm}\)

2. INFER the approach

• This is a right triangle problem with known hypotenuse and a relationship between the legs
• We need the Pythagorean theorem: \(\mathrm{a^2 + b^2 = c^2}\)
• Our equation will be: \(\mathrm{(shorter\,leg)^2 + (longer\,leg)^2 = (hypotenuse)^2}\)

3. Set up the Pythagorean equation

Substituting our expressions:

\(\mathrm{(x - 7)^2 + x^2 = 13^2}\)

4. SIMPLIFY through algebraic expansion

• Expand \(\mathrm{(x - 7)^2}\): \(\mathrm{x^2 - 14x + 49}\)
• Our equation becomes: \(\mathrm{(x^2 - 14x + 49) + x^2 = 169}\)
• Combine like terms: \(\mathrm{2x^2 - 14x + 49 = 169}\)
• Move everything to one side: \(\mathrm{2x^2 - 14x - 120 = 0}\)
• Divide by 2: \(\mathrm{x^2 - 7x - 60 = 0}\)

5. SIMPLIFY by factoring the quadratic

• We need two numbers that multiply to -60 and add to -7
• Those numbers are -12 and +5
• Factor: \(\mathrm{(x - 12)(x + 5) = 0}\)
• Solutions: \(\mathrm{x = 12}\) or \(\mathrm{x = -5}\)

6. APPLY CONSTRAINTS to select the valid answer

• Since x represents a length, it must be positive
• Therefore \(\mathrm{x = 12\,cm}\)

7. Verify our answer

• Longer leg = 12 cm, shorter leg = 12 - 7 = 5 cm
• Check: \(\mathrm{5^2 + 12^2 = 25 + 144 = 169}\) ✓
• And \(\mathrm{13^2 = 169}\) ✓

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might confuse which leg is longer and set up the relationship backwards, making the shorter leg = x and longer leg = \(\mathrm{(x + 7)}\). This leads to the wrong quadratic equation and ultimately an incorrect answer. Alternatively, they might solve correctly but answer with the shorter leg length instead of what the problem asks for.

This may lead them to select Choice A (5) - giving the shorter leg instead of the longer leg that the problem requests.

Second Most Common Error:

Poor SIMPLIFY execution: Students often make algebraic errors when expanding \(\mathrm{(x - 7)^2}\) or when factoring the quadratic equation. Common mistakes include sign errors in expansion or incorrect factoring, leading to wrong solutions.

This causes them to get stuck or arrive at incorrect values, often leading to guessing among the remaining choices.

The Bottom Line:

This problem tests both careful reading (to set up the relationship correctly) and solid algebra skills (expanding and factoring). Students need to track which leg is which throughout the entire solution process.