In the xy-plane, a circle has center at \((-3, 2)\) and passes through the point \((1, 2)\). What is the...

1. TRANSLATE the problem information

• Given information:
  • Circle center: (-3, 2)
  • Circle passes through point: (1, 2)
  • Need to find: diameter

• What "passes through" means: The point (1, 2) is on the circle

2. INFER the solution strategy

• To find diameter, we need radius first (since diameter = 2 × radius)
• Since (1, 2) is on the circle, the radius equals the distance from center (-3, 2) to point (1, 2)
• Use the distance formula to calculate this distance

3. SIMPLIFY using the distance formula

• Distance from (-3, 2) to (1, 2):

\(\mathrm{r = \sqrt{[(1 - (-3))^2 + (2 - 2)^2]}}\)

\(\mathrm{r = \sqrt{[(4)^2 + (0)^2]}}\)

\(\mathrm{r = \sqrt{16} = 4}\)

4. INFER the final calculation

• Since diameter = 2 × radius:

\(\mathrm{diameter = 2 \times 4 = 8}\)

Answer: C (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete INFER reasoning: Students correctly calculate the radius as 4, but forget that the problem asks for diameter, not radius.

They stop after finding \(\mathrm{r = 4}\) and select Choice A (4), missing the crucial final step of doubling the radius to get the diameter.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when handling the negative coordinate in the distance formula.

Common mistake: calculating (1 - (-3)) as 2 instead of 4, leading to \(\mathrm{r = \sqrt{4 + 0} = 2}\), then diameter = 4. This leads them to select Choice A (4).

The Bottom Line:

This problem tests whether students can connect multiple concepts: understanding what "passes through" means, applying the distance formula correctly, and remembering to convert radius to diameter. The key insight is recognizing that finding the diameter requires a two-step process.