In the figure, a rectangle has side lengths m = 6 and n = 8.A diagonal of the rectangle is...

1. TRANSLATE the problem information

Looking at the diagram and problem:

• Given information:
  • Rectangle ABCD with vertices labeled
  • Horizontal side (AB): \(\mathrm{n = 8}\)
  • Vertical side (AD): \(\mathrm{m = 6}\)
  • Diagonal from A to C is labeled \(\mathrm{d}\)

• What we need: An expression that gives the exact value of \(\mathrm{d}\) (not a numerical answer, but the expression itself)

2. INFER the geometric relationship

Here's the key insight: When you draw a diagonal across a rectangle, it creates a right triangle.

Why? Because all angles in a rectangle are \(\mathrm{90°}\). So:

• The diagonal AC becomes the hypotenuse
• The two sides of the rectangle (AB and BC) become the legs of the right triangle

Since we have a right triangle with:

• One leg = \(\mathrm{m = 6}\)
• Other leg = \(\mathrm{n = 8}\)
• Hypotenuse = \(\mathrm{d}\)

We can use the Pythagorean theorem!

3. TRANSLATE the Pythagorean theorem

For a right triangle: \(\mathrm{(leg_1)^2 + (leg_2)^2 = (hypotenuse)^2}\)

Substituting our values:

\(\mathrm{m^2 + n^2 = d^2}\)

\(\mathrm{6^2 + 8^2 = d^2}\)

4. Solve for d

To find \(\mathrm{d}\), take the square root of both sides:

\(\mathrm{d = \sqrt{m^2 + n^2}}\)

\(\mathrm{d = \sqrt{6^2 + 8^2}}\)

5. TRANSLATE to match answer choices

Looking at the options, we need the expression exactly as written:

Choice (D): \(\mathrm{\sqrt{6^2 + 8^2}}\)

This matches our expression exactly!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the diagonal creates a right triangle, so they don't see that the Pythagorean theorem applies. Instead, they think about the problem as "two sides combine to make a diagonal" and simply add the lengths together.

Their reasoning: "The diagonal goes across both dimensions, so I should add them: \(\mathrm{6 + 8}\)"

This leads them to select Choice A (\(\mathrm{6 + 8}\)).

Second Most Common Error:

Incomplete conceptual knowledge of Pythagorean theorem: Students remember that the Pythagorean theorem involves squares and square roots, but don't recall the exact form. They might think:

• "I need to multiply the sides somehow and take a square root"
• Or they confuse it with the geometric mean formula

This confusion may lead them to select Choice B (\(\mathrm{\sqrt{6 \cdot 8}}\)), which looks mathematical but uses multiplication instead of the sum of squares.

The Bottom Line:

This problem tests whether students can recognize a familiar formula (Pythagorean theorem) hidden in a geometric context (rectangle diagonal). The key breakthrough is INFERRING that a rectangle's diagonal forms a right triangle—without this insight, students resort to simpler but incorrect operations like addition or multiplication.