A rectangle has a perimeter of 40 centimeters. One of its sides measures 11 centimeters. What is the length, in...

1. TRANSLATE the problem information

• Given information:
  • Rectangle perimeter = 40 centimeters
  • One side length = 11 centimeters
  • Need to find: diagonal length

2. INFER what we need to find first

• To find a diagonal, we need both side lengths of the rectangle
• We have one side (11 cm) but need to find the other side
• We can use the perimeter to find the missing side

3. SIMPLIFY to find the unknown side length

• Perimeter formula: \(\mathrm{P = 2(l + w) = 40}\)
• Therefore: \(\mathrm{l + w = 20}\)
• If one side = 11 cm, then: other side = \(\mathrm{20 - 11 = 9}\) cm

4. INFER the geometric relationship

• The diagonal of a rectangle creates a right triangle
• The two sides (11 cm and 9 cm) are the legs
• The diagonal is the hypotenuse
• This means we can use the Pythagorean theorem

5. SIMPLIFY using the Pythagorean theorem

• \(\mathrm{d^2 = 11^2 + 9^2}\)
• \(\mathrm{d^2 = 121 + 81 = 202}\)
• \(\mathrm{d = \sqrt{202}}\)

Answer: (D) \(\mathrm{\sqrt{202}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find the other side length first before applying the Pythagorean theorem. They might try to use the perimeter directly in some formula or get confused about what information they actually have versus what they need.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Calculation errors in SIMPLIFY: Students correctly identify the approach but make arithmetic mistakes when computing \(\mathrm{11^2 + 9^2 = 121 + 81 = 202}\), or they might forget to take the square root at the end.

This may lead them to select an incorrect numerical choice or get a completely wrong value.

The Bottom Line:

This problem tests whether students can break down a multi-step geometry problem: first using perimeter to find missing information, then applying the Pythagorean theorem. The key insight is recognizing that you don't have all the information you need immediately, so you must work systematically through the given constraints.