The length of a rectangle's diagonal is 3sqrt(17), and the length of the rectangle's shorter side is 3. What is...

1. TRANSLATE the problem information

• Given information:
  • Rectangle's diagonal length = \(3\sqrt{17}\)
  • Rectangle's shorter side = \(3\)
  • Need to find: rectangle's longer side

• What this tells us: We have a rectangle where we know one side and the diagonal, and we need the other side.

2. INFER the geometric relationship

• Key insight: The diagonal of a rectangle creates a right triangle
• The two sides of the rectangle are the legs of this right triangle
• The diagonal is the hypotenuse
• This means we can use the Pythagorean theorem: \(\mathrm{a}^2 + \mathrm{b}^2 = \mathrm{c}^2\)

3. TRANSLATE into the Pythagorean theorem setup

• Let \(\mathrm{a}\) = shorter side = \(3\)
• Let \(\mathrm{b}\) = longer side (what we're solving for)
• Let \(\mathrm{c}\) = diagonal = \(3\sqrt{17}\)
• Equation: \(3^2 + \mathrm{b}^2 = (3\sqrt{17})^2\)

4. SIMPLIFY the equation

• First, calculate the right side:
  \((3\sqrt{17})^2 = 3^2 × (\sqrt{17})^2\)
  \(= 9 × 17\)
  \(= 153\) (use calculator)
• Now we have: \(9 + \mathrm{b}^2 = 153\)
• Subtract 9: \(\mathrm{b}^2 = 144\)
• Take the square root: \(\mathrm{b} = ±\sqrt{144} = ±12\)

5. APPLY CONSTRAINTS to select final answer

• Since we're looking for a length, which must be positive: \(\mathrm{b} = 12\)

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the rectangle's diagonal forms a right triangle with its sides.

They might try to use formulas for rectangle area or perimeter instead of connecting to the Pythagorean theorem. This leads to confusion and guessing since those formulas don't directly help find a missing side when given the diagonal.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \((3\sqrt{17})^2\).

They might incorrectly compute this as \(3\sqrt{17} × 3\sqrt{17} = 9\sqrt{17}\) instead of \(9 × 17 = 153\). This leads to an incorrect equation setup and a wrong final answer.

The Bottom Line:

This problem tests whether students can bridge rectangular geometry with right triangle relationships - a connection that isn't immediately obvious but becomes straightforward once recognized.