In the xy-plane, the coordinates of point P are \((2\sqrt{11}, -3\sqrt{11})\) and the coordinates of point Q are \((10\sqrt{11}, 12\sqrt{11})\)....

1. TRANSLATE the problem information

• Given information:
  • Point P has coordinates \((2\sqrt{11}, -3\sqrt{11})\)
  • Point Q has coordinates \((10\sqrt{11}, 12\sqrt{11})\)
  • Need to find the distance between these points

• What this tells us: We have two points in the xy-plane and need to use the distance formula.

2. INFER the approach

• Since we have coordinates of two points, we need the distance formula: \(\mathrm{d} = \sqrt{(\mathrm{x_2} - \mathrm{x_1})^2 + (\mathrm{y_2} - \mathrm{y_1})^2}\)
• First, we'll calculate the differences in x-coordinates and y-coordinates
• Then substitute these differences into the formula

3. SIMPLIFY to find coordinate differences

• Calculate \(\Delta\mathrm{x} = \mathrm{x_2} - \mathrm{x_1}\):
  \(\Delta\mathrm{x} = 10\sqrt{11} - 2\sqrt{11} = (10 - 2)\sqrt{11} = 8\sqrt{11}\)

• Calculate \(\Delta\mathrm{y} = \mathrm{y_2} - \mathrm{y_1}\):
  \(\Delta\mathrm{y} = 12\sqrt{11} - (-3\sqrt{11}) = 12\sqrt{11} + 3\sqrt{11} = (12 + 3)\sqrt{11} = 15\sqrt{11}\)

4. SIMPLIFY by substituting into the distance formula

• \(\mathrm{d} = \sqrt{(8\sqrt{11})^2 + (15\sqrt{11})^2}\)

• Expand the squared terms:
  \(\mathrm{d} = \sqrt{64 \times 11 + 225 \times 11}\)

• Factor out the common 11:
  \(\mathrm{d} = \sqrt{11(64 + 225)} = \sqrt{11 \times 289}\)

• Use properties of radicals:
  \(\mathrm{d} = \sqrt{11} \times \sqrt{289} = \sqrt{11} \times 17 = 17\sqrt{11}\)

Answer: \(17\sqrt{11}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make computational errors when handling the radical expressions, particularly when squaring terms like \((8\sqrt{11})^2\).

They might incorrectly compute \((8\sqrt{11})^2\) as \(8^2 \times \sqrt{11} = 64\sqrt{11}\) instead of \(8^2 \times (\sqrt{11})^2 = 64 \times 11 = 704\). This leads to expressions that can't be simplified properly, causing them to either get confused and guess or select an incorrect simplified form.

This may lead them to select Choice A \((13\sqrt{11})\) or get stuck and randomly select an answer.

Second Most Common Error:

Weak TRANSLATE reasoning: Students misidentify which coordinates belong to which point or make sign errors when calculating coordinate differences.

For example, they might calculate \(\Delta\mathrm{y}\) as \(12\sqrt{11} - 3\sqrt{11} = 9\sqrt{11}\) instead of \(12\sqrt{11} - (-3\sqrt{11}) = 15\sqrt{11}\), missing the negative sign in the y-coordinate of point P.

This leads to an incorrect calculation like \(\sqrt{(8\sqrt{11})^2 + (9\sqrt{11})^2} = \sqrt{11(64 + 81)} = \sqrt{11 \times 145}\), which doesn't yield any of the answer choices and causes confusion and guessing.

The Bottom Line:

This problem tests careful handling of radical expressions and attention to coordinate signs. Success requires systematic application of the distance formula combined with methodical algebraic simplification.