In the xy-plane, what is the distance between point R, with coordinates \((-2, 5)\), and point S, with coordinates \((10,...

1. TRANSLATE the problem information

• Given information:
  • Point R has coordinates \((-2, 5)\)
  • Point S has coordinates \((10, -1)\)
  • Need to find the distance between these points

2. INFER the approach

• This is a distance problem in the coordinate plane
• I need to use the distance formula: \(\mathrm{d} = \sqrt{(\mathrm{x_2} - \mathrm{x_1})^2 + (\mathrm{y_2} - \mathrm{y_1})^2}\)
• This formula comes from the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical distances between the points

3. TRANSLATE coordinates into the formula

• Let Point R = \((\mathrm{x_1}, \mathrm{y_1}) = (-2, 5)\)
• Let Point S = \((\mathrm{x_2}, \mathrm{y_2}) = (10, -1)\)

4. SIMPLIFY by calculating the coordinate differences

• Horizontal distance: \(\mathrm{x_2} - \mathrm{x_1} = 10 - (-2) = 10 + 2 = 12\)
• Vertical distance: \(\mathrm{y_2} - \mathrm{y_1} = -1 - 5 = -6\)

5. SIMPLIFY using the distance formula

• \(\mathrm{d} = \sqrt{(12)^2 + (-6)^2}\)
• \(\mathrm{d} = \sqrt{144 + 36}\)
• \(\mathrm{d} = \sqrt{180}\)

6. SIMPLIFY the radical

• Look for perfect square factors: \(180 = 36 \times 5\)
• Since \(36 = 6^2\), we have: \(\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}\)

Answer: B. \(6\sqrt{5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

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

For example, they might calculate \(\mathrm{x_2} - \mathrm{x_1} = 10 - (-2) = 10 - 2 = 8\) instead of \(10 + 2 = 12\). Or they might forget that \((-6)^2 = 36\), not \(-36\). These arithmetic errors lead to incorrect values under the square root, making it impossible to match any of the given answer choices. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly calculate \(\sqrt{180}\) but fail to simplify the radical.

They might leave their answer as \(\sqrt{180}\) and not recognize that this can be simplified to \(6\sqrt{5}\). Since \(\sqrt{180}\) is not among the answer choices, they may pick the closest-looking option or guess. Some might try to approximate \(\sqrt{180} \approx 13.4\) (use calculator) and mistakenly select Choice A (12) as the closest integer.

The Bottom Line:

This problem tests both computational accuracy and radical simplification skills. The distance formula itself is straightforward, but executing it correctly with negative coordinates and then recognizing how to factor and simplify \(\sqrt{180}\) requires careful attention to algebraic details.