In the xy-plane, a circle has a diameter with endpoints at \((-3, 5)\) and \((9, -7)\). What is the length...

1. TRANSLATE the problem information

• Given information:
  • Circle has diameter with endpoints at \((-3, 5)\) and \((9, -7)\)
  • Need to find the radius length
• What this tells us: We need to find the distance between these two points (which gives us the diameter), then divide by 2.

2. INFER the solution approach

• The distance between the endpoints gives us the diameter
• Once we have the diameter, radius = diameter ÷ 2
• We'll use the distance formula to find the distance between the two points

3. Apply the distance formula

• Distance between \((-3, 5)\) and \((9, -7)\):
  \(\mathrm{d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}\)
  \(\mathrm{d = \sqrt{(9-(-3))^2 + (-7-5)^2}}\)
  \(\mathrm{d = \sqrt{(12)^2 + (-12)^2}}\)
  \(\mathrm{d = \sqrt{144 + 144}}\)
  \(\mathrm{d = \sqrt{288}}\)

4. SIMPLIFY the radical

• \(\mathrm{\sqrt{288} = \sqrt{144 \times 2} = \sqrt{12^2 \times 2} = 12\sqrt{2}}\)
• So the diameter = \(\mathrm{12\sqrt{2}}\)

5. Find the radius

• Radius = diameter ÷ 2 = \(\mathrm{\frac{12\sqrt{2}}{2} = 6\sqrt{2}}\)

Answer: B (\(\mathrm{6\sqrt{2}}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students calculate the distance correctly as \(\mathrm{12\sqrt{2}}\), but then think this IS the radius rather than the diameter.

They forget that the distance between the endpoints of a diameter gives the diameter length, not the radius. This leads them to select Choice D (\(\mathrm{12\sqrt{2}}\)) instead of recognizing they need to divide by 2.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{\sqrt{288}}\) but struggle to simplify the radical properly.

They might leave it as \(\mathrm{\sqrt{288}}\) or incorrectly simplify it (perhaps trying \(\mathrm{\sqrt{288} = \sqrt{144} + \sqrt{144} = 12 + 12 = 24}\)), leading to confusion and potentially selecting Choice C (12) through incomplete radical work.

The Bottom Line:

This problem tests whether students truly understand the relationship between diameter and radius, not just the distance formula. The key insight is recognizing that finding the distance between diameter endpoints gives you the full diameter, which must then be halved to get the radius.