The height of a triangle is 6 centimeters. The base of the triangle is 3 times as long as the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Height = 6\ centimeters}\)
  • Base = "3 times as long as the height"
• What this tells us: \(\mathrm{Base = 3 \times 6 = 18\ centimeters}\)

2. INFER the solution strategy

• We have both dimensions needed for the triangle area formula
• Apply: \(\mathrm{Area = \frac{1}{2} \times base \times height}\)

3. SIMPLIFY the area calculation

• \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
• \(\mathrm{Area = \frac{1}{2} \times 18 \times 6}\)
• \(\mathrm{Area = \frac{1}{2} \times 108 = 54\ square\ centimeters}\)

Answer: D (54)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misunderstand "3 times as long as the height" and use the multiplier (3) directly in the area formula instead of the actual base length (18).

They calculate: \(\mathrm{Area = \frac{1}{2} \times 6 \times 3 = 9}\), or try \(\mathrm{Area = \frac{1}{2} \times height \times 3 = \frac{1}{2} \times 6 \times 3 = 9}\). Since 9 isn't an option, they might double it or make other adjustments, potentially leading them to select Choice B (27) thinking \(\mathrm{Area = 3 \times 9 = 27}\).

Second Most Common Error:

Missing conceptual knowledge of triangle area formula: Students forget the \(\mathrm{\frac{1}{2}}\) factor and calculate \(\mathrm{Area = base \times height = 18 \times 6 = 108}\). Since 108 isn't among the choices, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can accurately translate proportional language into mathematical relationships and then systematically apply the triangle area formula with all its components.