Triangle A has a base of 8 centimeters and a height of 9 centimeters. Triangle B has the same height...

1. TRANSLATE the problem information

• Given information:
  • Triangle A: base = 8 cm, height = 9 cm
  • Triangle B: same height as A (9 cm), area = \(\frac{3}{2}\) × area of A
  • Find: base of triangle B

2. SIMPLIFY to find triangle A's area

• Use the triangle area formula: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• Area of triangle A = \(\frac{1}{2} \times 8 \times 9 = 36\) square centimeters

3. TRANSLATE the area relationship for triangle B

• Area of triangle B = \(\frac{3}{2} \times \mathrm{area\ of\ triangle\ A}\)
• Area of triangle B = \(\frac{3}{2} \times 36 = 54\) square centimeters

4. SIMPLIFY to find triangle B's base

• Use the area formula with triangle B's known area and height:
• \(54 = \frac{1}{2} \times \mathrm{base} \times 9\)
• \(54 = \frac{9}{2} \times \mathrm{base}\)
• \(\mathrm{base} = 54 \times \frac{2}{9} = 12\) centimeters

Answer: B (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "3/2 times the area of triangle A" and think triangle B's base should be 3/2 times triangle A's base, completely bypassing the area calculation step.

They might calculate: base of B = \(\frac{3}{2} \times 8 = 12\), getting the right answer by luck, or they might get confused about what measurement is being scaled and make errors like base = \(8 \times \frac{2}{3} = \frac{16}{3}\), leading to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the final equation \(54 = \frac{9}{2} \times \mathrm{base}\) but make arithmetic errors when solving for the base, such as incorrectly calculating \(54 \div \frac{9}{2}\) or forgetting to multiply by the reciprocal.

This leads to wrong numerical answers that might make them select Choice A (6) if they divided incorrectly, or Choice C (16) if they made reciprocal errors.

The Bottom Line:

This problem tests whether students can work systematically through a multi-step area relationship rather than jumping to conclusions about which measurements are directly proportional.