In a circle with radius 13sqrt(37), a chord is at a perpendicular distance of 5sqrt(37) from the center. What is...

1. VISUALIZE the geometric setup

• Given information:
  • Circle radius: \(13\sqrt{37}\)
  • Perpendicular distance from center to chord: \(5\sqrt{37}\)
  • Need to find: chord length

• VISUALIZE by drawing a perpendicular line from the center to the chord - this creates a right triangle

2. INFER the key geometric relationship

• The perpendicular from center to chord bisects the chord (cuts it exactly in half)
• This creates a right triangle where:
  • Hypotenuse = radius = \(13\sqrt{37}\)
  • One leg = perpendicular distance = \(5\sqrt{37}\)
  • Other leg = half the chord length

3. INFER the solution approach

• Use Pythagorean theorem to find half the chord length
• Then double it for the full chord

4. SIMPLIFY using Pythagorean theorem

• Set up: \((\text{half chord})^2 + (5\sqrt{37})^2 = (13\sqrt{37})^2\)
• Expand: \((\text{half chord})^2 + 25 \cdot 37 = 169 \cdot 37\)
• Solve for half chord: \((\text{half chord})^2 = 169 \cdot 37 - 25 \cdot 37 = 144 \cdot 37\)
• Take square root: \(\text{half chord} = \sqrt{144 \cdot 37} = \sqrt{144} \cdot \sqrt{37} = 12\sqrt{37}\)

5. Complete the solution

• Full chord = \(2 \times (\text{half chord}) = 2 \times 12\sqrt{37} = 24\sqrt{37}\)

Answer: C (\(24\sqrt{37}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Students fail to draw the perpendicular from center to chord, missing the crucial right triangle setup.

Without this visualization, they might try to use the given measurements directly in formulas like circumference or area, or attempt to work with the chord and radius without the perpendicular relationship. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the Pythagorean theorem correctly but make algebraic errors when handling \(\sqrt{144 \cdot 37}\).

They might calculate \(\sqrt{144 \cdot 37}\) as \(\sqrt{144} + \sqrt{37}\) instead of \(\sqrt{144} \cdot \sqrt{37}\), or fail to recognize that \(\sqrt{144} = 12\). This could lead them to select Choice A (\(12\sqrt{37}\)) if they forget to double the half-chord result.

The Bottom Line:

This problem tests whether students can visualize the geometric relationship between radius, chord, and perpendicular distance. The key insight is recognizing that the perpendicular from center creates a right triangle with the chord bisected.