A triangle has an area of 84 square centimeters and a base of 12 centimeters. What is the height, in...

1. TRANSLATE the problem information

• Given information:
  • Area = 84 square centimeters
  • Base = 12 centimeters
  • Need to find: height

2. INFER the approach

• We need the triangle area formula to connect these three quantities
• Since we have area and base, we can solve for height directly

3. TRANSLATE into mathematical equation

• Triangle area formula: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• Substitute our values: \(84 = \frac{1}{2} \times 12 \times \mathrm{height}\)

4. SIMPLIFY to solve for height

• First simplify the right side: \(84 = 6 \times \mathrm{height}\)
• Divide both sides by 6: \(\mathrm{height} = 84 \div 6 = 14\)

Answer: C (14 centimeters)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students forget the factor of 1/2 in the triangle area formula and instead use \(\mathrm{Area} = \mathrm{base} \times \mathrm{height}\).

This leads them to set up: \(84 = 12 \times \mathrm{height}\), giving \(\mathrm{height} = 84 \div 12 = 7\).

This may lead them to select Choice B (7).

Second Most Common Error:

Weak SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors when dividing 84 by 6.

Some might incorrectly calculate 84 ÷ 6 as 6 or double-check incorrectly and choose other values.

This can cause confusion leading to guessing among the remaining choices.

The Bottom Line:

Triangle area problems are straightforward once you remember the complete formula including the 1/2 factor, but that factor is the most commonly forgotten component that leads to systematic errors.