In the xy-plane, points R and S are endpoints of a diameter of a circle. If the center of the...

1. TRANSLATE the problem information

• Given information:
  • Center of circle: \((3, -1)\)
  • Point R (one endpoint of diameter): \((7, 2)\)
  • Need to find point S (other endpoint of diameter)

• What this tells us: Since R and S are endpoints of a diameter, the center must be exactly halfway between them (the midpoint).

2. INFER the mathematical approach

• Key insight: If the center is the midpoint of diameter RS, then we can use the midpoint formula in reverse
• Strategy: Set up equations where center = midpoint of R and S, then solve for S's coordinates

3. TRANSLATE the midpoint relationship into equations

• Using midpoint formula: Center \((3, -1)\) = midpoint of \mathrm{R}(7, 2)\) and \mathrm{S}(x, y)\)
• This gives us:
  • For x-coordinate: \(3 = \frac{7 + x}{2}\)
  • For y-coordinate: \(-1 = \frac{2 + y}{2}\)

4. SIMPLIFY to find the x-coordinate

• From \(3 = \frac{7 + x}{2}\):
  • Multiply both sides by 2: \(6 = 7 + x\)
  • Subtract 7 from both sides: \(x = -1\)

5. SIMPLIFY to find the y-coordinate

• From \(-1 = \frac{2 + y}{2}\):
  • Multiply both sides by 2: \(-2 = 2 + y\)
  • Subtract 2 from both sides: \(y = -4\)

Answer: A \((-1, -4)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misunderstand the geometric relationship and think the center coordinates ARE point S, rather than recognizing that the center is the midpoint BETWEEN R and S.

This leads them to immediately select Choice B \((3, -1)\) without doing any calculations.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the midpoint equations but make sign errors during algebraic manipulation, particularly when dealing with negative numbers in the y-coordinate calculation.

For example, they might solve \(-1 = \frac{2 + y}{2}\) as \(-2 = 2 + y\), then incorrectly conclude \(y = 0\) or \(y = 4\), leading them toward incorrect choices or confusion.

The Bottom Line:

This problem requires students to work backwards from a midpoint relationship - they must recognize that knowing the midpoint and one endpoint allows them to find the other endpoint. The key insight is translating "center of circle" and "diameter endpoints" into the mathematical concept of midpoint.