In the coordinate plane, the distance between points P and Q is 50 units. The x-coordinate of P is 14...

1. TRANSLATE the problem information

• Given information:
  • Distance between points P and Q is 50 units
  • x-coordinate of P is 14 units greater than x-coordinate of Q
  • Need: positive difference between y-coordinates

• This tells us we have a right triangle where the hypotenuse is the distance between points

2. INFER the approach

• This is a distance formula problem in disguise
• We can think of the coordinate differences as legs of a right triangle, with the distance as the hypotenuse
• We know one leg (x-difference = 14) and the hypotenuse (distance = 50), need to find the other leg (y-difference)

3. TRANSLATE into the distance formula

• Distance formula: \(\mathrm{d^2 = (\Delta x)^2 + (\Delta y)^2}\)
• Substitute known values: \(\mathrm{50^2 = 14^2 + (\Delta y)^2}\)

4. SIMPLIFY to solve for the y-difference

• Calculate: \(\mathrm{2500 = 196 + (\Delta y)^2}\)
• Subtract: \(\mathrm{(\Delta y)^2 = 2500 - 196 = 2304}\)
• Take square root: \(\mathrm{\Delta y = \pm\sqrt{2304} = \pm48}\)

5. APPLY CONSTRAINTS to select final answer

• Since we want the positive difference between y-coordinates: \(\mathrm{\Delta y = 48}\)

Answer: C (48)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students see "x-coordinate of P is 14 units greater" and think the answer is simply 14, not recognizing this as a distance formula problem requiring the Pythagorean theorem.

They focus only on the x-coordinate difference and miss that this difference, combined with the y-coordinate difference, creates a right triangle with the 50-unit distance as the hypotenuse.

This may lead them to select Choice A (14).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make computational errors in the algebra, particularly when calculating \(\mathrm{2500 - 196 = 2304}\) or finding \(\mathrm{\sqrt{2304}}\).

Common arithmetic mistakes might lead to incorrect intermediate values, causing confusion about which answer choice matches their work.

This causes them to get stuck or select an incorrect choice through computational error.

The Bottom Line:

This problem tests whether students can recognize coordinate geometry as an application of the Pythagorean theorem, rather than just memorizing the distance formula as an isolated concept.