In the coordinate plane, point P has coordinates \((0, 0)\) and point Q has coordinates \((15, 25)\). What is the...

1. TRANSLATE the problem information

• Given information:
  • Point P is at \((0, 0)\)
  • Point Q is at \((15, 25)\)
  • Need to find distance between these points

2. INFER the approach

• Distance problems in coordinate plane require the distance formula
• We have both points' coordinates, so we can substitute directly
• Formula: \(\mathrm{d} = \sqrt{(\mathrm{x_2}-\mathrm{x_1})^2 + (\mathrm{y_2}-\mathrm{y_1})^2}\)

3. TRANSLATE coordinates into the distance formula

• P\((0, 0)\) means \(\mathrm{x_1} = 0, \mathrm{y_1} = 0\)
• Q\((15, 25)\) means \(\mathrm{x_2} = 15, \mathrm{y_2} = 25\)
• \(\mathrm{d} = \sqrt{(15-0)^2 + (25-0)^2} = \sqrt{15^2 + 25^2}\)

4. Calculate and SIMPLIFY

• \(15^2 = 225\) and \(25^2 = 625\) (use calculator)
• \(\mathrm{d} = \sqrt{225 + 625} = \sqrt{850}\)
• Factor 850 to simplify: \(850 = 25 \times 34 = 5^2 \times 34\)
• Therefore: \(\mathrm{d} = \sqrt{5^2 \times 34} = 5\sqrt{34}\)

Answer: B (\(5\sqrt{34}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\sqrt{850}\) but fail to factor it properly or don't recognize that 25 is a perfect square factor.

They might leave their answer as \(\sqrt{850}\) or incorrectly factor it (like thinking \(850 = 85 \times 10\)), leading to confusion when they don't see \(\sqrt{850}\) among the choices. This leads to guessing between the radical answer choices.

Second Most Common Error:

Conceptual gap with the distance formula: Students might try to find distance by simply adding coordinates (\(15 + 25 = 40\)) or subtracting them, forgetting that distance requires the Pythagorean-based distance formula.

This may lead them to select Choice D (40).

The Bottom Line:

The key challenge is recognizing that coordinate plane distance problems always require the distance formula, then executing the radical simplification correctly. Most errors come from either skipping the distance formula entirely or getting lost in the arithmetic and radical simplification steps.