Triangle R has an area of 80 square centimeters (cm^2). Square S has side lengths of 4 ~cm. What is...

1. TRANSLATE the problem information

• Given information:
  • Triangle R has an area of \(80 \text{ cm}^2\)
  • Square S has side lengths of \(4 \text{ cm}\)
• What we need to find: Total area of both shapes

2. INFER the solution approach

• Since we have the triangle's area already, we just need it as-is
• For the square, we need to calculate its area using the side length
• Then we'll add both areas together for the total

3. Calculate the square's area

Process Skill: SIMPLIFY using the area formula:

\(\mathrm{Area} = \mathrm{side}^2\)

Area of square S = \(4^2 = 16 \text{ cm}^2\)

4. Find the total area

Process Skill: SIMPLIFY by adding:

\(\mathrm{Total\,area} = 80 + 16 = 96 \text{ cm}^2\)

Answer: D. 96

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion: Students might use the square's side length \(4 \text{ cm}\) directly instead of calculating its area \(16 \text{ cm}^2\).

This happens when students don't fully apply the area formula and just grab the given number. They would calculate: \(80 + 4 = 84 \text{ cm}^2\).

This may lead them to select Choice C (84).

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating \(4^2\) or when adding the final numbers.

For example, they might calculate \(4^2 = 8\) instead of 16, leading to \(80 + 8 = 88 \text{ cm}^2\) (though this specific result isn't among the choices, similar calculation errors could lead to other incorrect selections).

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can distinguish between a measurement (side length) and a calculated property (area), then perform accurate arithmetic. The key insight is recognizing that having a "side length of 4 cm" means you need to square it to get the area.