A rectangle measures 56 centimeters by 112 centimeters. When a diagonal is drawn, it divides the rectangle into two congruent...

1. TRANSLATE the problem information

• Given information:
  • Rectangle dimensions: \(56\text{ cm} \times 112\text{ cm}\)
  • Diagonal divides rectangle into two congruent right triangles
  • Need to find area of one triangle

2. INFER the triangle's dimensions

• Key insight: When a diagonal cuts through a rectangle, each resulting triangle has:
  • Base = one side of the rectangle \(56\text{ cm}\)
  • Height = the other side of the rectangle \(112\text{ cm}\)
• The diagonal becomes the hypotenuse, but we don't need it for area calculation

3. SIMPLIFY using the triangle area formula

• \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• \(\mathrm{Area} = \frac{1}{2} \times 56 \times 112\)
• Calculate: \(56 \times 112 = 6{,}272\) (use calculator)
• \(\mathrm{Area} = \frac{1}{2} \times 6{,}272 = 3{,}136\text{ square centimeters}\)

Answer: C (3,136)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think they need to find the diagonal length first, then try to use more complex triangle formulas.

They might attempt to find the diagonal using the Pythagorean theorem: \(\mathrm{d}^2 = 56^2 + 112^2\), then try to use this with other triangle formulas. This unnecessary complication leads to confusion and often incorrect calculations.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make calculation errors.

They correctly identify \(\mathrm{Area} = \frac{1}{2} \times 56 \times 112\), but miscalculate the multiplication or forget to divide by 2. For example, calculating \(56 \times 112\) incorrectly or forgetting the \(\frac{1}{2}\) factor entirely.

This may lead them to select Choice D (6,272) if they forget to divide by 2.

The Bottom Line:

This problem tests whether students can see the simple relationship between a rectangle and the triangles formed by its diagonal, without getting distracted by unnecessary calculations involving the diagonal itself.