In the xy-plane, line segment JK is transformed by a dilation with a center of dilation at the origin and...

1. TRANSLATE the problem information

• Given information:
  • Original line segment JK with endpoints \(\mathrm{J(2, 5)}\) and \(\mathrm{K(4, 9)}\)
  • Dilation from origin with scale factor 3
  • Need to find slope of transformed segment \(\mathrm{J'K'}\)

2. INFER the most efficient approach

• Key insight: You have two solution paths here:
  • Fast path: Recognize that dilations preserve slopes (angles remain unchanged)
  • Calculation path: Find new coordinates, then calculate slope directly
• Both work, but the conceptual approach is faster and less error-prone

3a. SIMPLIFY using the conceptual approach (recommended)

• Since dilation preserves angles, it preserves slopes
• Calculate original slope: \(\mathrm{m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2}\)
• The dilated segment has the same slope: 2

3b. SIMPLIFY using the calculation approach (alternative)

• Apply dilation formula: \(\mathrm{(x, y) \rightarrow (3x, 3y)}\)
  • \(\mathrm{J(2, 5) \rightarrow J'(6, 15)}\)
  • \(\mathrm{K(4, 9) \rightarrow K'(12, 27)}\)
• Calculate new slope: \(\mathrm{m = \frac{27 - 15}{12 - 6} = \frac{12}{6} = 2}\)

Answer: B. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that dilations preserve slopes, leading to unnecessary and potentially error-prone coordinate calculations.

Students often jump straight into finding the new coordinates without realizing there's a more direct conceptual approach. While the calculation method works, it involves more steps where arithmetic mistakes can occur. Some students might make errors when multiplying coordinates by the scale factor or when computing the final slope with larger numbers.

This typically doesn't lead to a specific wrong answer choice but makes the problem more time-consuming and increases chances of calculation errors.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during coordinate transformation or slope calculation.

Students might correctly identify that they need to find new coordinates but make mistakes like:

• Forgetting to multiply both coordinates by 3
• Arithmetic errors in the slope calculation (especially with the larger numbers in the dilated coordinates)
• Mixing up the order in the slope formula

These calculation errors could lead them to select any of the incorrect answer choices depending on the specific arithmetic mistake made.

The Bottom Line:

This problem tests whether students understand that similarity transformations (like dilations) preserve angle relationships. Students who grasp this concept can solve it in seconds, while those who don't must rely on potentially error-prone calculations.