The area of a right triangle is 66 square feet. The length of one leg of the triangle is 10...

1. TRANSLATE the problem information

• Given information:
  • Area of right triangle = 66 square feet
  • One leg = 10 more than twice the other leg
  • Need to find the shorter leg
• Setting up variables: Let \(\mathrm{x}\) = shorter leg, so other leg = \(\mathrm{2x + 10}\)

2. INFER the solution approach

• Since we know the area and have expressions for both legs, we can use the right triangle area formula
• This will create one equation with one unknown that we can solve

3. TRANSLATE the area relationship into an equation

• Area formula: \(\mathrm{A} = \frac{1}{2} \times \mathrm{leg_1} \times \mathrm{leg_2}\)
• Substitute known values: \(66 = \frac{1}{2} \times \mathrm{x} \times (\mathrm{2x + 10})\)

4. SIMPLIFY the equation step by step

• Multiply both sides by 2: \(132 = \mathrm{x}(\mathrm{2x + 10})\)
• Distribute: \(132 = \mathrm{2x^2 + 10x}\)
• Rearrange to standard form: \(\mathrm{2x^2 + 10x - 132} = 0\)
• Divide by 2: \(\mathrm{x^2 + 5x - 66} = 0\)

5. SIMPLIFY by factoring the quadratic

• Look for two numbers that multiply to -66 and add to 5
• Those numbers are 11 and -6: \((11)(-6) = -66\) and \(11 + (-6) = 5\)
• Factor: \((\mathrm{x + 11})(\mathrm{x - 6}) = 0\)
• Solutions: \(\mathrm{x} = -11\) or \(\mathrm{x} = 6\)

6. APPLY CONSTRAINTS to select the valid answer

• Since length cannot be negative, \(\mathrm{x} = 6\) feet

Answer: C) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "10 more than twice the other leg" and setting up the wrong algebraic relationship. Students might write the longer leg as \(\mathrm{x + 10 + 2x} = \mathrm{3x + 10}\), or confuse which leg is shorter and write \(\mathrm{2x + 10} = \mathrm{x}\) (thinking the shorter leg is the complicated expression).

This leads to a completely different quadratic equation with different solutions, causing them to select Choice A (4) or Choice B (5) depending on their specific translation error.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic mistakes when expanding \(\mathrm{x}(\mathrm{2x + 10})\) or errors in factoring the quadratic equation. Common mistakes include forgetting to distribute properly (getting \(\mathrm{x^2 + 10x}\) instead of \(\mathrm{2x^2 + 10x}\)) or factoring incorrectly.

This may lead them to select Choice D (8) or another incorrect answer based on their computational errors.

The Bottom Line:

This problem requires careful translation of the verbal relationship into algebra, followed by systematic equation solving. Students who rush through the setup phase or make careless algebraic errors will struggle to reach the correct answer.