Point P is at \((2, 4)\) and point Q is at \((9, 14)\) on the coordinate plane. What is the...

1. TRANSLATE the problem information

• Given information:
  • Point P is at (2, 4)
  • Point Q is at (9, 14)
  • Need to find the distance between them

• This tells us we have:
  • \(\mathrm{x_1 = 2, y_1 = 4}\) (coordinates of point P)
  • \(\mathrm{x_2 = 9, y_2 = 14}\) (coordinates of point Q)

2. INFER the approach needed

• Since we need distance between two points on a coordinate plane, we must use the distance formula
• The distance formula requires the differences in x-coordinates and y-coordinates

3. SIMPLIFY by calculating coordinate differences

• \(\Delta\mathrm{x = x_2 - x_1 = 9 - 2 = 7}\)
• \(\Delta\mathrm{y = y_2 - y_1 = 14 - 4 = 10}\)

4. SIMPLIFY by applying the distance formula

• \(\mathrm{d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}\)
• \(\mathrm{d = \sqrt{(7)^2 + (10)^2}}\)
• \(\mathrm{d = \sqrt{49 + 100}}\)
• \(\mathrm{d = \sqrt{149}}\)

Answer: B (\(\sqrt{149}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors when finding coordinate differences or when squaring numbers.

For example, they might calculate \(\Delta\mathrm{x = 9 - 2 = 7}\) correctly, but then compute \(\Delta\mathrm{y = 14 - 4 = 12}\) instead of 10. This leads to \(\mathrm{d = \sqrt{7^2 + 12^2}}\)
\(\mathrm{= \sqrt{49 + 144}}\)
\(\mathrm{= \sqrt{193}}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY reasoning: Students forget to square the coordinate differences before adding them in the distance formula.

They might compute \(\mathrm{d = \sqrt{7 + 10}}\)
\(\mathrm{= \sqrt{17}}\), which also doesn't match any given choice. This fundamental misunderstanding of the distance formula leads them to abandon systematic solution and guess.

The Bottom Line:

This problem tests careful execution of a familiar formula more than conceptual understanding. Success depends on methodical calculation and recognizing that \(\sqrt{149}\) is already in its simplest radical form.