The length of a rectangle's diagonal is 5sqrt(17), and the length of the rectangle's shorter side is 5. What is...

1. TRANSLATE the problem information

• Given information:
  • Rectangle diagonal length: \(5\sqrt{17}\)
  • Shorter side length: 5
  • Need to find: longer side length

2. INFER the geometric relationship

• A rectangle's diagonal divides it into two congruent right triangles
• In each right triangle:
  • Hypotenuse = diagonal = \(5\sqrt{17}\)
  • One leg = shorter side = 5
  • Other leg = longer side = unknown
• This is a Pythagorean theorem problem: \(\mathrm{a}^2 + \mathrm{b}^2 = \mathrm{c}^2\)

3. SIMPLIFY by setting up and solving the equation

• Substitute into Pythagorean theorem:
  \(5^2 + \mathrm{b}^2 = (5\sqrt{17})^2\)

• SIMPLIFY the right side:
  \((5\sqrt{17})^2 = 5^2 \times (\sqrt{17})^2 = 25 \times 17 = 425\)

• So: \(25 + \mathrm{b}^2 = 425\)

• SIMPLIFY to isolate b²:
  \(\mathrm{b}^2 = 425 - 25 = 400\)

• SIMPLIFY by taking the square root:
  \(\mathrm{b} = \sqrt{400} = 20\)

Answer: B. 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the diagonal creates right triangles requiring the Pythagorean theorem. Instead, they attempt direct division: \(5\sqrt{17} \div 5 = \sqrt{17}\).

This may lead them to select Choice A (\(\sqrt{17}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the Pythagorean theorem but make calculation errors. They might incorrectly expand \((5\sqrt{17})^2\) or forget to take the square root of \(\mathrm{b}^2 = 400\).

Forgetting the final square root step leads them to select Choice D (400).

The Bottom Line:

This problem requires students to visualize the geometric relationship between a rectangle and its diagonal, then execute multi-step algebra carefully. The key insight is recognizing that "diagonal" means "hypotenuse of a right triangle."