Points P and Q are located at coordinates \((3, 5)\) and \((15, 12)\), respectively, on a coordinate plane. What is...

1. TRANSLATE the problem information

• Given information:
  • Point P is at coordinates \((3, 5)\)
  • Point Q is at coordinates \((15, 12)\)
  • Need to find the distance between them

2. INFER the approach needed

• This is asking for distance between two points on a coordinate plane
• The distance formula is required: \(\mathrm{d} = \sqrt{(\mathrm{x}_2-\mathrm{x}_1)^2 + (\mathrm{y}_2-\mathrm{y}_1)^2}\)
• I need to identify which coordinates are \((\mathrm{x}_1,\mathrm{y}_1)\) and which are \((\mathrm{x}_2,\mathrm{y}_2)\)

3. Set up the distance formula

• Let \(\mathrm{P}(3, 5)\) be \((\mathrm{x}_1, \mathrm{y}_1)\) and \(\mathrm{Q}(15, 12)\) be \((\mathrm{x}_2, \mathrm{y}_2)\)
• Distance = \(\sqrt{(15-3)^2 + (12-5)^2}\)

4. SIMPLIFY through the calculation

• Calculate the differences: \((15-3) = 12\) and \((12-5) = 7\)
• Square each difference: \(12^2 = 144\) and \(7^2 = 49\)
• Add them: \(144 + 49 = 193\)
• Take the square root: \(\sqrt{193}\)

Answer: C \((\sqrt{193})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the multi-step calculation process.

They might miscalculate one of the squares (like getting \(7^2 = 48\) instead of 49) or make addition errors. Some students also forget to take the final square root step, calculating \(144 + 49 = 193\) and selecting Choice D (193) instead of \(\sqrt{193}\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread or confuse the coordinate values.

Instead of carefully using \((3,5)\) and \((15,12)\), they might misread the coordinates or switch values around, leading to calculations like \(\sqrt{(10-3)^2 + (8-5)^2} = \sqrt{49 + 9} = \sqrt{58}\). This causes them to select Choice A \((\sqrt{58})\).

Third Common Error:

Missing conceptual knowledge: Students don't remember the distance formula properly.

Some students try to find distance by simply adding the coordinate differences: \((15-3) + (12-5) = 12 + 7 = 19\), leading them to select Choice B (19).

The Bottom Line:

This problem tests both formula recall and careful arithmetic execution. Students who know the distance formula but rush through calculations are particularly vulnerable to errors.