A right triangle has legs of lengths x and x + 1, and hypotenuse x + 2, where x is...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs of length x and x + 1
  • Hypotenuse of length x + 2
  • x is positive
• What this tells us: We can use the Pythagorean theorem since we have a right triangle

2. INFER the approach

• Since we have a right triangle, the Pythagorean theorem applies: \(\mathrm{a^2 + b^2 = c^2}\)
• Set up the equation with the two legs as a and b, and hypotenuse as c
• This gives us: \(\mathrm{x^2 + (x + 1)^2 = (x + 2)^2}\)

3. SIMPLIFY by expanding the squared expressions

• Expand \(\mathrm{(x + 1)^2: x^2 + 2x + 1}\)
• Expand \(\mathrm{(x + 2)^2: x^2 + 4x + 4}\)
• Equation becomes: \(\mathrm{x^2 + x^2 + 2x + 1 = x^2 + 4x + 4}\)

4. SIMPLIFY by combining and rearranging terms

• Combine like terms on left: \(\mathrm{2x^2 + 2x + 1 = x^2 + 4x + 4}\)
• Subtract \(\mathrm{x^2}\) from both sides: \(\mathrm{x^2 + 2x + 1 = 4x + 4}\)
• Move all terms to one side: \(\mathrm{x^2 + 2x + 1 - 4x - 4 = 0}\)
• Simplify: \(\mathrm{x^2 - 2x - 3 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Look for two numbers that multiply to -3 and add to -2
• Those numbers are -3 and +1: \(\mathrm{(x - 3)(x + 1) = 0}\)
• Solutions: \(\mathrm{x = 3}\) or \(\mathrm{x = -1}\)

6. APPLY CONSTRAINTS to select final answer

• Since x must be positive: \(\mathrm{x = 3}\)

Answer: (C) 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors during the expansion or combining of terms. For example, they might expand \(\mathrm{(x + 1)^2}\) incorrectly as \(\mathrm{x^2 + 1}\) (forgetting the middle term 2x), or make sign errors when moving terms to one side of the equation.

This leads to an incorrect quadratic equation, which when solved gives wrong values that may match other answer choices. This could lead them to select Choice (A) (1), Choice (B) (2), or Choice (D) (4).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might incorrectly set up the initial equation by confusing which sides are legs versus hypotenuse, or by misunderstanding the relationship between x, x + 1, and x + 2.

This fundamental setup error propagates through the entire solution, leading to confusion and likely guessing among the answer choices.

The Bottom Line:

This problem requires careful algebraic manipulation over multiple steps. Students who rush through the expansion and simplification phases, or who don't systematically check their algebra, are likely to make computational errors that lead to wrong answers.