Triangle F and triangle G are similar triangles. The area of triangle G is 1,024 times the area of triangle...

1. TRANSLATE the problem information

• Given information:
  • Triangle F and triangle G are similar
  • \(\mathrm{Area\ of\ G} = 1,024 \times \mathrm{Area\ of\ F}\)
  • Need to find: How many times larger is perimeter of G compared to F?

2. INFER the key relationship for similar figures

• Since the triangles are similar, there's a scaling factor k between corresponding lengths
• For similar figures: if linear dimensions scale by k, then areas scale by \(\mathrm{k}^2\)
• This means: \(\frac{\mathrm{Area\ of\ G}}{\mathrm{Area\ of\ F}} = \mathrm{k}^2\)

3. TRANSLATE the area relationship into an equation

• We know: \(\mathrm{Area\ of\ G} = 1,024 \times \mathrm{Area\ of\ F}\)
• So: \(\frac{\mathrm{Area\ of\ G}}{\mathrm{Area\ of\ F}} = 1,024\)
• Therefore: \(\mathrm{k}^2 = 1,024\)

4. SIMPLIFY to find the scaling factor

• Take the square root of both sides: \(\mathrm{k} = \sqrt{1,024}\)
• To calculate \(\sqrt{1,024}\): recognize that \(1,024 = 2^{10}\)
• So: \(\mathrm{k} = \sqrt{2^{10}} = 2^5 = 32\)

5. INFER the perimeter relationship

• For similar figures, perimeters scale by the same factor as linear dimensions
• Therefore: \(\frac{\mathrm{Perimeter\ of\ G}}{\mathrm{Perimeter\ of\ F}} = \mathrm{k} = 32\)

Answer: A (32)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the relationship between area scaling and linear scaling for similar figures.

Many students see that the area is 1,024 times larger and incorrectly assume the perimeter is also 1,024 times larger. They fail to realize that areas scale by \(\mathrm{k}^2\) while linear dimensions (including perimeter) scale by k. This leads them to select Choice D (1,024).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up \(\mathrm{k}^2 = 1,024\) correctly but make calculation errors finding the square root.

Some students might calculate \(\sqrt{1,024}\) incorrectly, perhaps getting confused and calculating \(\sqrt{512} = 16\sqrt{2} \approx 22.6\), leading them toward Choice A (32) by coincidence, or making other computational mistakes that lead to Choice B (64) if they mistakenly think \(64^2 = 1,024\).

The Bottom Line:

This problem tests whether students truly understand the scaling relationships in similar figures - that linear dimensions and areas don't scale by the same factor. The key insight is recognizing that if areas differ by factor \(\mathrm{k}^2\), then linear dimensions (like perimeter) differ by factor k.