In the xy-plane, a quadrilateral FGHJ is transformed by a dilation centered at the origin to create a similar quadrilateral...

1. TRANSLATE the problem information

• Given information:
  • Quadrilateral FGHJ dilated from origin to create QRST
  • \(\mathrm{F(6, 10)}\) corresponds to \(\mathrm{Q(9, 15)}\)
  • \(\mathrm{G(8, 2)}\) corresponds to R (unknown)
  • Need: sum of R's coordinates

2. INFER the solution strategy

• Since this is a dilation centered at origin, all coordinates scale by the same factor k
• Strategy: Find k using known corresponding vertices F and Q, then apply k to G to find R

3. SIMPLIFY to find the scale factor

• From \(\mathrm{F(6, 10) \rightarrow Q(9, 15)}\):
  • x-coordinate: \(\mathrm{6k = 9}\), so \(\mathrm{k = \frac{9}{6} = \frac{3}{2} = 1.5}\)
  • y-coordinate: \(\mathrm{10k = 15}\), so \(\mathrm{k = \frac{15}{10} = 1.5}\) ✓

4. SIMPLIFY to find vertex R

• Apply \(\mathrm{k = 1.5}\) to \(\mathrm{G(8, 2)}\):
  • \(\mathrm{R_x = 8 \times 1.5 = 12}\)
  • \(\mathrm{R_y = 2 \times 1.5 = 3}\)
  • Therefore \(\mathrm{R = (12, 3)}\)

5. Calculate final answer

• Sum of R's coordinates = \(\mathrm{12 + 3 = 15}\)

Answer: B (15)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find the scale factor first using the given corresponding vertices.

Instead, they might try to directly find some relationship between G and R without establishing the dilation factor, leading to random guessing or attempting to use incorrect geometric relationships. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need a scale factor but make arithmetic errors when calculating it.

For example, they might get \(\mathrm{k = \frac{2}{3}}\) instead of \(\mathrm{\frac{3}{2}}\), leading to \(\mathrm{R = (\frac{16}{3}, \frac{4}{3})}\) and \(\mathrm{sum \approx 6.7}\), which doesn't match any answer choice. This causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that dilation creates a predictable scaling relationship that can be determined from any pair of corresponding points, then systematically applied to find unknown vertices.