In the xy-plane, a line segment has endpoints at \(\mathrm{A(-2, 1)}\) and \(\mathrm{B(4, -6)}\). What is the length of the...

1. TRANSLATE the problem information

• Given information:
  • Point A has coordinates (-2, 1)
  • Point B has coordinates (4, -6)
  • Need to find the length of line segment AB

• What this tells us: We need to find the distance between two points in the coordinate plane

2. INFER the approach

• To find distance between two points, we use the distance formula
• The distance formula: \(\mathrm{d} = \sqrt{(\mathrm{x_2} - \mathrm{x_1})^2 + (\mathrm{y_2} - \mathrm{y_1})^2}\)
• We'll substitute our coordinates and calculate step by step

3. TRANSLATE coordinates into the formula

• From A(-2, 1): \(\mathrm{x_1} = -2\), \(\mathrm{y_1} = 1\)
• From B(4, -6): \(\mathrm{x_2} = 4\), \(\mathrm{y_2} = -6\)
• Set up: \(\mathrm{d} = \sqrt{(4 - (-2))^2 + ((-6) - 1)^2}\)

4. SIMPLIFY the coordinate differences

• \(\mathrm{x_2} - \mathrm{x_1} = 4 - (-2) = 4 + 2 = 6\)
• \(\mathrm{y_2} - \mathrm{y_1} = -6 - 1 = -7\)
• Now we have: \(\mathrm{d} = \sqrt{6^2 + (-7)^2}\)

5. SIMPLIFY to final answer

• \(6^2 = 36\)
• \((-7)^2 = 49\)
• \(\mathrm{d} = \sqrt{36 + 49} = \sqrt{85}\)

Answer: C. \(\sqrt{85}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when calculating coordinate differences, especially with negative coordinates.

Common mistake: Calculating \(\mathrm{x_2} - \mathrm{x_1} = 4 - (-2) = 4 - 2 = 2\) instead of \(4 + 2 = 6\), or \(\mathrm{y_2} - \mathrm{y_1} = -6 - 1 = -5\) instead of \(-7\). These errors lead to wrong values under the radical, such as \(\sqrt{4 + 25} = \sqrt{29}\) or \(\sqrt{36 + 25} = \sqrt{61}\).

This may lead them to select Choice B (\(\sqrt{61}\)) if they get the y-difference wrong.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or incorrectly apply the distance formula.

Some might try to find distance by simply subtracting coordinates or adding them, rather than using the proper geometric relationship. Others might forget to square the differences or forget the square root entirely.

This leads to confusion and guessing among the radical answer choices.

The Bottom Line:

The key challenge is careful arithmetic with signed numbers while applying a memorized formula. Success requires both knowing the distance formula and executing the calculations precisely, especially when dealing with negative coordinates.