A rhombus ABCD has diagonals AC and BD that intersect perpendicularly at point P. The diagonals bisect each other at...

1. TRANSLATE the problem information

• Given information:
  • Rhombus ABCD with perpendicular diagonals intersecting at P
  • Diagonals bisect each other at P
  • \(\mathrm{AC = 600~mm}\)
  • Perimeter = 2,000 mm
• Need to find: Length of diagonal BD

2. INFER what the rhombus properties tell us

• Since diagonals bisect each other: \(\mathrm{AP = PC = 600/2 = 300~mm}\)
• Let \(\mathrm{BP = PD = x~mm}\), so \(\mathrm{BD = 2x~mm}\) (we'll solve for x)
• Since all sides of a rhombus are equal: each side = \(\mathrm{2,000 ÷ 4 = 500~mm}\)

3. INFER the geometric relationship to use

• The perpendicular diagonals create four congruent right triangles
• Focus on triangle APB where:
  • AB = 500 mm (side of rhombus)
  • AP = 300 mm (half of diagonal AC)
  • BP = x mm (half of diagonal BD)
• Since the diagonals are perpendicular, we can use Pythagorean theorem

4. SIMPLIFY using Pythagorean theorem

• \(\mathrm{AB^2 = AP^2 + BP^2}\)
• \(\mathrm{500^2 = 300^2 + x^2}\)
• \(\mathrm{250,000 = 90,000 + x^2}\)
• \(\mathrm{x^2 = 160,000}\)
• \(\mathrm{x = 400~mm}\)

5. TRANSLATE back to find the full diagonal

• \(\mathrm{BD = 2x = 2(400) = 800~mm}\)

Answer: D. 800

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to use the Pythagorean theorem on the right triangles formed by the perpendicular diagonals. Instead, they might try to use diagonal formulas or get confused about how to connect the given information. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the Pythagorean equation correctly (\(\mathrm{500^2 = 300^2 + x^2}\)) but make arithmetic errors when solving for x. They might calculate \(\mathrm{500^2 - 300^2}\) incorrectly or make errors finding the square root of 160,000. This could lead them to select Choice B (500) if they confuse the side length with the diagonal length.

The Bottom Line:

This problem requires recognizing that a rhombus's perpendicular diagonals create a framework for applying the Pythagorean theorem—students must see the geometric relationship, not just memorize formulas.