Question:In the coordinate plane, the endpoints of a diameter of a circle are \(\mathrm{A(-5, 7)}\) and \(\mathrm{B(7, -11)}\). The radius...

1. TRANSLATE the problem information

• Given information:
  • Diameter endpoints: \(\mathrm{A(-5, 7)}\) and \(\mathrm{B(7, -11)}\)
  • Need radius in form \(\mathrm{3\sqrt{d}}\)
• What this tells us: We need to find the distance between these points (diameter), then find half of that (radius)

2. INFER the approach

• Since we have diameter endpoints, use the distance formula to find diameter length
• Then divide by 2 to get radius
• Finally, match the result to form \(\mathrm{3\sqrt{d}}\)

3. Apply distance formula

• Distance \(\mathrm{AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}\)
• \(\mathrm{\Delta x = 7-(-5) = 12}\), \(\mathrm{\Delta y = -11-7 = -18}\)
• Distance = \(\mathrm{\sqrt{(12)^2 + (-18)^2}}\)
  = \(\mathrm{\sqrt{144 + 324}}\)
  = \(\mathrm{\sqrt{468}}\)

4. SIMPLIFY the radical

• Factor 468: \(\mathrm{468 = 4 \times 117 = 4 \times 9 \times 13 = 36 \times 13}\)
• \(\mathrm{\sqrt{468} = \sqrt{36 \times 13}}\)
  = \(\mathrm{\sqrt{36} \times \sqrt{13}}\)
  = \(\mathrm{6\sqrt{13}}\)
• So diameter = \(\mathrm{6\sqrt{13}}\)

5. Find the radius

• INFER that radius = diameter ÷ 2
• Radius = \(\mathrm{(6\sqrt{13}) \div 2}\)
  = \(\mathrm{3\sqrt{13}}\)

6. TRANSLATE to find d

• We have radius = \(\mathrm{3\sqrt{13}}\)
• Comparing to form \(\mathrm{3\sqrt{d}}\): \(\mathrm{d = 13}\)

Answer: 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Forgetting that radius is half the diameter

Students correctly find the diameter as \(\mathrm{6\sqrt{13}}\), but then give this as their final answer, thinking \(\mathrm{d = 13}\) corresponds to \(\mathrm{6\sqrt{13}}\) rather than \(\mathrm{3\sqrt{13}}\). They miss the crucial step of dividing by 2.

This leads to confusion when trying to match \(\mathrm{6\sqrt{13}}\) to the form \(\mathrm{3\sqrt{d}}\).

Second Most Common Error:

Poor SIMPLIFY execution: Incorrectly simplifying \(\mathrm{\sqrt{468}}\)

Students might factor incorrectly (like \(\mathrm{468 = 4 \times 117}\) but missing the further factorization) or make arithmetic errors when computing \(\mathrm{12^2 + (-18)^2 = 144 + 324}\). This leads to wrong radical expressions that can't be simplified properly.

This causes them to get stuck and guess.

The Bottom Line:

This problem requires strong coordinate geometry skills combined with careful radical simplification. The key insight is remembering that finding the diameter is only halfway to the solution - you must then find the radius.