A rectangle is inscribed in a circle such that all four vertices of the rectangle lie on the circumference of...

1. TRANSLATE the ratio information

• Given information:
  • Rectangle inscribed in circle (all vertices on circumference)
  • Sides in ratio \(3:4\)
  • Area = \(432\) square units

• TRANSLATE the ratio: Let the sides be \(3k\) and \(4k\) for some positive value \(k\)

2. SIMPLIFY to find the scale factor k

• Set up the area equation: \(\mathrm{Area} = \mathrm{length} \times \mathrm{width}\)
• \((3k)(4k) = 432\)
• \(12k^2 = 432\)
• \(k^2 = 36\)
• \(k = 6\)

3. Calculate the actual side lengths

• Shorter side: \(3k = 3(6) = 18\) units
• Longer side: \(4k = 4(6) = 24\) units

4. INFER the key geometric relationship

• For any rectangle inscribed in a circle, the diagonal of the rectangle equals the diameter of the circle
• This is because the diagonal spans the full width of the circle

5. SIMPLIFY using the Pythagorean theorem

• \(\mathrm{Diagonal} = \sqrt{18^2 + 24^2}\)
• \(= \sqrt{324 + 576}\)
• \(= \sqrt{900} = 30\)

Answer: C) 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing that the diagonal of an inscribed rectangle equals the diameter of the circumscribing circle.

Students might correctly find the side lengths (18 and 24) but then get stuck because they don't recognize this crucial geometric relationship. They might try to use the circumference formula or guess at relationships between the rectangle's dimensions and the circle's radius, leading to confusion and random guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Making calculation errors when applying the Pythagorean theorem.

Students might correctly set up \(\sqrt{18^2 + 24^2}\) but then calculate \(18^2 = 324\) and \(24^2 = 576\) incorrectly, or make errors when adding \(324 + 576 = 900\). Common wrong calculations might yield \(\sqrt{841} = 29\), leading them to select Choice B (26) as the closest option.

The Bottom Line:

This problem requires connecting rectangle properties with circle geometry - the key insight that makes everything click is recognizing that an inscribed rectangle's diagonal spans the circle's diameter. Without this connection, students have all the pieces but can't finish the solution.