A circle in the xy-plane has its center at the point \(\mathrm{(2, -1)}\). The point \(\mathrm{(7, 11)}\) lies on the...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((2, -1)\)
  • Point on circle: \((7, 11)\)
  • Need to find: radius

2. INFER the mathematical approach

• Key insight: The radius of a circle is the distance from the center to any point on the circle
• Since we know both the center and a point on the circle, we can find the radius by calculating the distance between these two points
• Tool needed: Distance formula

3. TRANSLATE into distance formula setup

• Set up the distance formula: \(\mathrm{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\)
• Identify coordinates:
  • \((\mathrm{x_1, y_1}) = (2, -1)\) [center]
  • \((\mathrm{x_2, y_2}) = (7, 11)\) [point on circle]

4. SIMPLIFY through the calculation

• Calculate the differences:
  • \(\mathrm{x_2 - x_1 = 7 - 2 = 5}\)
  • \(\mathrm{y_2 - y_1 = 11 - (-1) = 11 + 1 = 12}\)

• Substitute into formula:
  • \(\mathrm{d = \sqrt{5^2 + 12^2}}\)
  • \(\mathrm{d = \sqrt{25 + 144}}\)
  • \(\mathrm{d = \sqrt{169}}\)
  • \(\mathrm{d = 13}\)

Answer: C) 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't immediately recognize that "radius" means "distance from center to point on circle"

Instead, they might try to use other circle formulas (like circumference or area) or get confused about what the problem is actually asking for. This leads to confusion and guessing rather than systematic problem-solving.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic mistakes in the multi-step calculation

Common calculation errors include:

• Getting \(\mathrm{11 - (-1) = 10}\) instead of 12
• Miscalculating \(\mathrm{5^2 + 12^2}\) as something other than 169
• Not recognizing that \(\mathrm{\sqrt{169} = 13}\)

These errors might lead them to select Choice A (5) if they only calculated one coordinate difference, or Choice E (169) if they forgot to take the final square root.

The Bottom Line:

This problem tests whether students can connect the geometric concept of radius with the algebraic tool of the distance formula. Success requires both conceptual understanding and careful arithmetic execution.