Question:A right triangle has legs of length x units and \(\mathrm{(x - 15)}\) units, where x gt 15. The square...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs \(\mathrm{x}\) and \(\mathrm{(x - 15)}\) units
  • \(\mathrm{x \gt 15}\) (ensures positive leg lengths)
  • Square of hypotenuse = 377
  • Find the value of x
• What this tells us: We can use the Pythagorean theorem since we know it's a right triangle

2. INFER the mathematical approach

• Since we have a right triangle, the Pythagorean theorem applies: \(\mathrm{a^2 + b^2 = c^2}\)
• Our legs are \(\mathrm{x}\) and \(\mathrm{(x - 15)}\), and \(\mathrm{c^2 = 377}\)
• This gives us: \(\mathrm{x^2 + (x - 15)^2 = 377}\)

3. SIMPLIFY the equation through algebraic expansion

• Expand \(\mathrm{(x - 15)^2}\): \(\mathrm{(x - 15)^2 = x^2 - 30x + 225}\)
• Substitute: \(\mathrm{x^2 + x^2 - 30x + 225 = 377}\)
• Combine like terms: \(\mathrm{2x^2 - 30x + 225 = 377}\)
• Move all terms to left: \(\mathrm{2x^2 - 30x - 152 = 0}\)
• Divide by 2: \(\mathrm{x^2 - 15x - 76 = 0}\)

4. SIMPLIFY using the quadratic formula

• For \(\mathrm{x^2 - 15x - 76 = 0}\), we have \(\mathrm{a = 1, b = -15, c = -76}\)
• \(\mathrm{x = \frac{15 ± \sqrt{225 + 304}}{2}}\)
  \(\mathrm{= \frac{15 ± \sqrt{529}}{2}}\)
  \(\mathrm{= \frac{15 ± 23}{2}}\)
• This gives \(\mathrm{x = 19}\) or \(\mathrm{x = -4}\)

5. APPLY CONSTRAINTS to select the final answer

• Since \(\mathrm{x \gt 15}\) (given constraint), we reject \(\mathrm{x = -4}\)
• Therefore \(\mathrm{x = 19}\)

Answer: B (19)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make algebraic errors when expanding \(\mathrm{(x - 15)^2}\) or combining terms, leading to an incorrect quadratic equation. A common mistake is expanding as \(\mathrm{x^2 - 15x + 225}\) instead of \(\mathrm{x^2 - 30x + 225}\), forgetting to double the middle term.

This algebraic error propagates through the quadratic formula, producing wrong solutions that may lead them to select Choice A (4) or cause confusion leading to guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret which sides are the legs versus the hypotenuse, or incorrectly set up the Pythagorean theorem. Some students might think one of the given expressions represents the hypotenuse rather than a leg.

This fundamental setup error leads to a completely different equation, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem combines geometric understanding with algebraic manipulation. Success requires both recognizing the Pythagorean theorem setup AND executing multi-step algebraic simplification accurately. The constraint application is straightforward once you get the right solutions, but getting there requires solid algebra skills.