A circle is drawn in the xy-plane. The center of the circle is at the point \((-\frac{1}{2}, 4)\). If one...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((-1/2, 4)\)
  • One endpoint of diameter: \((5/2, 13/3)\)
  • Need to find the other endpoint

2. INFER the geometric relationship

• Key insight: The center of any circle is the midpoint of every diameter
• This means we can use the midpoint formula to find the missing endpoint
• Strategy: Set up the midpoint formula with the center as the midpoint and solve for the unknown coordinates

3. TRANSLATE the relationship into algebra

• Let the unknown endpoint be \((x, y)\)
• Using midpoint formula: Center = \((\mathrm{endpoint}_1 + \mathrm{endpoint}_2)/2\) for each coordinate)
• Set up equations:
  • For x: \(-1/2 = (5/2 + x)/2\)
  • For y: \(4 = (13/3 + y)/2\)

4. SIMPLIFY to find the x-coordinate

• Start with: \(-1/2 = (5/2 + x)/2\)
• Multiply both sides by 2: \(-1 = 5/2 + x\)
• Solve for x: \(x = -1 - 5/2\)
• Convert to common denominator: \(x = -2/2 - 5/2 = -7/2\)

5. SIMPLIFY to find the y-coordinate

• Start with: \(4 = (13/3 + y)/2\)
• Multiply both sides by 2: \(8 = 13/3 + y\)
• Solve for y: \(y = 8 - 13/3\)
• Convert to common denominator: \(y = 24/3 - 13/3 = 11/3\)

Answer: A \((-7/2, 11/3)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the center is the midpoint of the diameter, so they attempt to use other circle properties like the distance formula or try to find the radius first.

Without this key geometric insight, they may calculate distances or try complex approaches that don't lead to a solution. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the midpoint equations correctly but make algebraic errors when working with fractions, particularly when converting between mixed numbers and improper fractions or finding common denominators.

For example, they might incorrectly calculate \(y = 8 - 13/3\) as \(8 - 4⅓ = 3⅔\) instead of properly converting \(8 = 24/3\). This may lead them to select Choice D \((-1/2, -5/3)\) or another incorrect answer.

The Bottom Line:

This problem tests whether students can connect geometric properties (center as midpoint) to algebraic tools (midpoint formula). The geometric insight is crucial - without it, students will struggle to even begin the problem systematically.