A circle in the xy-plane has center \((3, 5)\). The line y = 7 is tangent to this circle. An...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((3, 5)\)
  • Tangent line: \(\mathrm{y = 7}\)
  • Circle equation form: \((\mathrm{x} - 3)^2 + (\mathrm{y} - 5)^2 = \mathrm{r}^2\)
  • Need to find: radius \(\mathrm{r}\)

2. INFER the key relationship

• When a line is tangent to a circle, it touches the circle at exactly one point
• The radius to this tangent point is perpendicular to the tangent line
• Therefore: radius = perpendicular distance from center to tangent line

3. SIMPLIFY the distance calculation

• Center is at \((3, 5)\), tangent line is \(\mathrm{y = 7}\)
• Since \(\mathrm{y = 7}\) is horizontal, I need the vertical distance
• Distance = \(|\mathrm{y\text{-coordinate of center}} - \mathrm{y\text{-value of line}}|\)
• Distance = \(|5 - 7| = |-2| = 2\)

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the tangent line property to distance calculation. They might try to substitute points or use the circle equation algebraically, getting confused about how to use the tangent condition. This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about tangent lines: Students remember that tangent means "touching at one point" but don't realize this means the radius equals the perpendicular distance to the line. They might try to find intersection points algebraically, leading to unnecessary complex calculations and potential errors.

The Bottom Line:

This problem tests whether students can translate the geometric meaning of "tangent" into a distance calculation. The key insight is recognizing that tangency gives us the radius directly through perpendicular distance.