The area A of a trapezoid can be represented by the expression \(\mathrm{h(b + 4)}\), where h is the height...

1. TRANSLATE the problem information

• Given information:
  • Area of trapezoid: \(\mathrm{A = h(b + 4)}\)
  • h = height
  • b = shorter base length
  • Need to find: expression for longer base

• We also know the standard trapezoid area formula: \(\mathrm{A = \frac{1}{2}(b_1 + b_2)h}\)

2. INFER the approach

• Since both expressions represent the same area of the same trapezoid, we can set them equal
• This will allow us to solve for the longer base (\(\mathrm{b_2}\))
• Strategy: Set \(\mathrm{h(b + 4) = \frac{1}{2}(b + b_2)h}\) and solve for \(\mathrm{b_2}\)

3. SIMPLIFY to set up the equation

Set the two area expressions equal:

\(\mathrm{\frac{1}{2}(b + b_2)h = h(b + 4)}\)

4. SIMPLIFY by dividing both sides by h

\(\mathrm{\frac{1}{2}(b + b_2) = b + 4}\)

5. SIMPLIFY by multiplying both sides by 2

\(\mathrm{b + b_2 = 2b + 8}\)

6. SIMPLIFY by isolating b₂

Subtract b from both sides:

\(\mathrm{b_2 = 2b + 8 - b = b + 8}\)

Answer: D (b + 8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that the given expression \(\mathrm{h(b + 4)}\) represents the same area as the standard formula, so they need to be set equal to each other.

Without this key insight, students might think the answer is simply one of the terms already visible in the given expression, leading them to select Choice C (b + 4) thinking this must be the longer base since it appears in the area expression.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation but make algebraic errors during the solving process.

For example, when they have \(\mathrm{b + b_2 = 2b + 8}\), they might incorrectly subtract to get \(\mathrm{b_2 = b + 4}\) instead of \(\mathrm{b_2 = b + 8}\). This may lead them to select Choice C (b + 4).

The Bottom Line:

This problem requires students to connect two different representations of the same geometric quantity and use algebraic manipulation to find the relationship between the bases. The key insight is recognizing that both expressions must be equal since they represent the same area.