A circle in the xy-plane has a diameter with endpoints \((2, 4)\) and \((2, 14)\). An equation of this circle...

1. TRANSLATE the problem information

• Given information:
  • Circle equation: \((x - 2)^2 + (y - 9)^2 = r^2\)
  • Diameter endpoints: \((2, 4)\) and \((2, 14)\)
  • Need to find: value of r
• What this tells us: The equation is in standard form with center at \((2, 9)\)

2. INFER the relationship between given information

• Key insight: Since diameter endpoints lie on the circle, the radius equals the distance from center to either endpoint
• Strategy: Use distance formula from center to one diameter endpoint

3. SIMPLIFY using distance formula

• From center \((2, 9)\) to endpoint \((2, 4)\):
  \(r = \sqrt{(2-2)^2 + (9-4)^2}\)
  \(r = \sqrt{0^2 + 5^2}\)
  \(r = \sqrt{25} = 5\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students calculate the length of the entire diameter instead of recognizing they need the radius (half the diameter).

They might find the distance between endpoints \((2, 4)\) and \((2, 14)\):
\(\mathrm{Distance} = \sqrt{(2-2)^2 + (14-4)^2}\)
\(= \sqrt{0^2 + 10^2}\)
\(= 10\)

Since this gives diameter \(= 10\), they incorrectly conclude \(r = 10\) instead of \(r = 5\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misidentify the center of the circle by not properly reading the standard form equation.

They might think the center is at \((2, 4)\) or \((2, 14)\) instead of \((2, 9)\), leading to incorrect distance calculations and wrong radius values.

The Bottom Line:

This problem tests whether students understand the relationship between diameter and radius, and whether they can properly interpret the standard form of a circle equation to identify the center.