A triangle has a base length of 56 centimeters and a height of 112 centimeters. What is the area, in...

1. TRANSLATE the problem information

• Given information:
  • Base length: \(\mathrm{b = 56}\) centimeters
  • Height: \(\mathrm{h = 112}\) centimeters
• Find: Area of the triangle in square centimeters

2. INFER the mathematical approach

• This is a triangle area problem, so we need the triangle area formula
• The formula is: \(\mathrm{A = \frac{1}{2}bh}\)
• We have both base and height values, so we can substitute directly

3. SIMPLIFY by substituting and calculating

• \(\mathrm{A = \frac{1}{2}bh}\)
• \(\mathrm{A = \frac{1}{2} \times 56 \times 112}\)
• \(\mathrm{A = \frac{1}{2} \times 6{,}272}\)
• \(\mathrm{A = 3{,}136}\)

Answer: C. 3,136

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor INFER reasoning: Students recognize they need to use base and height but don't recall or apply the correct triangle area formula. Instead, they might add the base and height together, thinking area involves combining the two measurements.

This leads them to calculate: \(\mathrm{56 + 112 = 168}\)

This may lead them to select Choice A (168)

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify the formula \(\mathrm{A = \frac{1}{2}bh}\) but make calculation errors. They might forget the ½ factor entirely and just multiply base times height.

This leads them to calculate: \(\mathrm{56 \times 112 = 6{,}272}\)

This may lead them to select Choice D (6,272)

The Bottom Line:

Triangle area problems seem simple, but students often either forget the specific formula or make arithmetic errors with the ½ factor. The key is systematically applying \(\mathrm{A = \frac{1}{2}bh}\) and carefully executing the multiplication.