A carpenter is constructing a right triangular brace for a wooden frame. One leg of the triangle is 3 centimeters...

1. TRANSLATE the problem information

• Given information:
  • Right triangular brace
  • One leg is 3 cm longer than the other leg
  • Area = 54 square cm
  • Need to find: length of longer leg

• Let \(\mathrm{x}\) = length of shorter leg (cm)
• Then \(\mathrm{x + 3}\) = length of longer leg (cm)

2. TRANSLATE the area relationship

• For a right triangle: \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
• The two legs serve as base and height
• So: \(\mathrm{\frac{1}{2} \times x \times (x + 3) = 54}\)

3. SIMPLIFY to solve for x

• Multiply both sides by 2: \(\mathrm{x(x + 3) = 108}\)
• Expand: \(\mathrm{x^2 + 3x = 108}\)
• Rearrange: \(\mathrm{x^2 + 3x - 108 = 0}\)
• Factor: \(\mathrm{(x + 12)(x - 9) = 0}\)
• Solutions: \(\mathrm{x = -12}\) or \(\mathrm{x = 9}\)

4. APPLY CONSTRAINTS to select final answer

• Since length must be positive in real-world context: \(\mathrm{x = 9}\) cm
• Therefore, longer leg = \(\mathrm{x + 3 = 9 + 3 = 12}\) cm

Answer: C) 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct area equation, particularly with the relationship between the legs. They might write the area as \(\mathrm{\frac{1}{2} \times x \times 3}\) instead of \(\mathrm{\frac{1}{2} \times x \times (x + 3)}\), thinking the "3 cm longer" means one leg is just 3 cm rather than understanding it's a relative measurement.

This leads them to solve \(\mathrm{\frac{1}{2} \times x \times 3 = 54}\), getting \(\mathrm{x = 36}\), and then adding 3 to get 39 cm. Since this isn't among the choices, this causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the quadratic equation but make errors in factoring \(\mathrm{x^2 + 3x - 108 = 0}\). They might factor incorrectly or use the quadratic formula but make arithmetic mistakes, leading to wrong values for x.

This may lead them to select Choice A (9) if they find \(\mathrm{x = 9}\) but forget to add 3 for the longer leg, or other incorrect choices based on their calculation errors.

The Bottom Line:

The key challenge is correctly translating the relative relationship between the legs into the area formula, then executing the algebraic solution without losing track of what the question is actually asking for (the longer leg, not the shorter one).