What is the diameter of the circle in the xy-plane with equation \((\mathrm{x} - 5)^2 + (\mathrm{y} - 3)^2 =...

1. TRANSLATE the equation into recognizable form

• Given equation: \((x - 5)^2 + (y - 3)^2 = 16\)
• This matches the standard form of a circle: \((x - h)^2 + (y - k)^2 = r^2\)
• By comparison: \(r^2 = 16\)

2. INFER what we need to find the diameter

• The question asks for diameter, but we first need to find the radius
• To find diameter, we'll use: \(\mathrm{diameter} = 2 \times \mathrm{radius}\)

3. SIMPLIFY to find the radius

• From \(r^2 = 16\), take the square root of both sides
• \(r = \pm\sqrt{16} = \pm4\)

4. APPLY CONSTRAINTS to select the valid radius

• Since radius represents a length, it must be positive
• Therefore: \(r = 4\)

5. Calculate the final answer

• \(\mathrm{Diameter} = 2 \times \mathrm{radius} = 2 \times 4 = 8\)

Answer: B. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly find \(r = 4\) but stop there, thinking they've answered the question.

They see "4" in the answer choices and immediately select it without realizing the question asks for diameter, not radius. The radius calculation feels like the main mathematical work, so it's natural to think that's the final answer.

This leads them to select Choice A (4).

Second Most Common Error:

Conceptual confusion about standard form: Students recognize the equation format but misinterpret what \(r^2\) represents in the context of the question.

They see \(r^2 = 16\) and think "the problem is asking about the circle's size, and 16 seems like the right number" without understanding that \(r^2\) is the square of the radius, not the diameter.

This may lead them to select Choice C (16).

The Bottom Line:

This problem tests whether students can distinguish between radius and diameter after extracting information from standard form. The mathematical steps are straightforward, but the conceptual leap from "find the radius" to "use radius to find diameter" is where most students stumble.