Question: A trapezoid has an area equal to x^2 square centimeters. The lengths of its two bases are x +...

1. TRANSLATE the problem information

• Given information:
  • Area = \(\mathrm{x^2}\) square centimeters
  • Base 1 length = \(\mathrm{x + 9}\) centimeters
  • Base 2 length = \(\mathrm{x - 1}\) centimeters
  • Height = \(\mathrm{x - 3}\) centimeters
• What this tells us: We need to use the trapezoid area formula to create an equation

2. INFER the solution approach

• Since we know the area formula and all dimensions in terms of x, we can set up one equation with one unknown
• Strategy: Use \(\mathrm{A = \frac{1}{2}(b_1 + b_2)h}\) and solve for x

3. Set up the equation using the trapezoid area formula

\(\mathrm{x^2 = \frac{1}{2}((x + 9) + (x - 1))(x - 3)}\)

4. SIMPLIFY the right side step by step

• First, combine the bases: \(\mathrm{(x + 9) + (x - 1) = 2x + 8}\)
• So: \(\mathrm{x^2 = \frac{1}{2}(2x + 8)(x - 3)}\)
• Factor out the 2: \(\mathrm{x^2 = (x + 4)(x - 3)}\)
• Expand: \(\mathrm{x^2 = x^2 + x - 12}\)

5. SIMPLIFY to solve for x

• Subtract \(\mathrm{x^2}\) from both sides: \(\mathrm{0 = x - 12}\)
• Therefore: \(\mathrm{x = 12}\)

6. APPLY CONSTRAINTS to verify the solution

• Check geometric validity: Height = \(\mathrm{x - 3 = 12 - 3 = 9 \gt 0}\) ✓
• Verification: Area = \(\mathrm{\frac{1}{2}(21 + 11)(9) = 144 = 12^2}\) ✓

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x + 4)(x - 3)}\), often getting \(\mathrm{x^2 - 3x + 4x + 12 = x^2 + x + 12}\) instead of \(\mathrm{x^2 + x - 12}\).

This leads to the equation \(\mathrm{x^2 = x^2 + x + 12}\), which gives \(\mathrm{-12 = x}\) instead of \(\mathrm{x = 12}\). This creates confusion since a negative value doesn't make geometric sense, leading to guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students incorrectly set up the area formula by forgetting the \(\mathrm{\frac{1}{2}}\) factor, writing \(\mathrm{x^2 = (2x + 8)(x - 3)}\) directly.

This leads to \(\mathrm{x^2 = 2x^2 + 2x - 24}\), giving \(\mathrm{x^2 + 2x - 24 = 0}\). Using the quadratic formula yields messy non-integer solutions, causing students to doubt their work and potentially guess.

The Bottom Line:

This problem requires careful algebraic manipulation where small errors compound quickly. Students must maintain precision through multiple steps of expansion and simplification while remembering to check that their final answer makes geometric sense.