A triangular banner has area T, in square inches, given by \(\mathrm{T = \frac{x}{2}(x + 16)}\), where x is the...

1. TRANSLATE the problem information

• Given information:
  • Triangle area: \(\mathrm{T} = \frac{\mathrm{x}}{2}(\mathrm{x} + 16)\)
  • \(\mathrm{x}\) = length of the base
• What we need: expression for the height

2. INFER the solution approach

• The key insight: We know the standard triangle area formula, so we can match it with the given formula
• Strategy: Use \(\mathrm{A} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\) and match it with the given expression

3. SIMPLIFY by matching the formulas

• Standard formula with our base: \(\mathrm{T} = \frac{1}{2} \times \mathrm{x} \times \mathrm{height} = \frac{\mathrm{x}}{2} \times \mathrm{height}\)
• Given formula: \(\mathrm{T} = \frac{\mathrm{x}}{2}(\mathrm{x} + 16)\)
• Since both equal T: \(\frac{\mathrm{x}}{2} \times \mathrm{height} = \frac{\mathrm{x}}{2}(\mathrm{x} + 16)\)

4. SIMPLIFY to isolate height

• Divide both sides by \(\frac{\mathrm{x}}{2}\): \(\mathrm{height} = \mathrm{x} + 16\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that they need to use the standard triangle area formula to solve this problem.

Students might look at the given expression \(\mathrm{T} = \frac{\mathrm{x}}{2}(\mathrm{x} + 16)\) and try to identify height by just looking at the terms, thinking "maybe height is x" or "maybe it's x/2" without systematically using the area formula. This leads to random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when matching the formulas.

Students might correctly identify that they need to match \(\mathrm{T} = \frac{\mathrm{x}}{2} \times \mathrm{height}\) with \(\mathrm{T} = \frac{\mathrm{x}}{2}(\mathrm{x} + 16)\), but then make errors like thinking the height equals the entire expression \(\frac{\mathrm{x}}{2}(\mathrm{x} + 16)\) rather than just the \((\mathrm{x} + 16)\) part. This may lead them to select Choice D \(\frac{\mathrm{x}(\mathrm{x} + 16)}{2}\).

The Bottom Line:

This problem tests whether students can work backwards from a given area formula to identify individual components, requiring them to connect the given expression with the fundamental triangle area relationship.