In a rectangle with length L, width w, and diagonal d, the relationship d^2 = L^2 + w^2 holds. What...

1. INFER the solution strategy

• We have the equation \(\mathrm{d^2 = L^2 + w^2}\) and need to find \(\mathrm{L}\)
• To isolate \(\mathrm{L}\), we need to first isolate \(\mathrm{L^2}\) by moving \(\mathrm{w^2}\) to the other side
• Then we'll take the square root to find \(\mathrm{L}\)

2. SIMPLIFY by isolating L²

• Starting equation: \(\mathrm{d^2 = L^2 + w^2}\)
• Subtract \(\mathrm{w^2}\) from both sides: \(\mathrm{d^2 - w^2 = L^2}\)

3. SIMPLIFY by taking the square root

• From \(\mathrm{L^2 = d^2 - w^2}\), take the square root of both sides
• \(\mathrm{L = \sqrt{d^2 - w^2}}\)
• Since \(\mathrm{L}\) represents length, we take the positive square root

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic sign errors and add \(\mathrm{w^2}\) instead of subtracting it.

Starting with \(\mathrm{d^2 = L^2 + w^2}\), they incorrectly think they need to "move \(\mathrm{w^2}\) to the other side" and write \(\mathrm{d^2 + w^2 = L^2}\). Taking the square root gives \(\mathrm{L = \sqrt{d^2 + w^2}}\).

This leads them to select Choice A (\(\mathrm{\sqrt{d^2 + w^2}}\))

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly isolate \(\mathrm{L^2 = d^2 - w^2}\) but forget to take the square root as the final step.

They stop at \(\mathrm{L^2 = d^2 - w^2}\) and select this as their answer, thinking they've solved for \(\mathrm{L}\) when they've only solved for \(\mathrm{L^2}\).

This leads them to select Choice D (\(\mathrm{d^2 - w^2}\))

The Bottom Line:

This problem tests careful algebraic manipulation. The key insight is recognizing that solving for \(\mathrm{L}\) is a two-step process: first isolate \(\mathrm{L^2}\) by subtracting \(\mathrm{w^2}\), then complete the solution by taking the square root.