Each side of equilateral triangle S is multiplied by a scale factor of k to create equilateral triangle T. The...

1. TRANSLATE the problem information

• Given information:
  • Triangle S is scaled by factor k to create triangle T
  • Triangle T has longer sides than triangle S
• What this tells us: We need k such that \(\mathrm{k \times (side\ of\ S) \gt (side\ of\ S)}\)

2. INFER the mathematical relationship

• Since scaling multiplies all dimensions by k, if the new triangle is larger, then k must be greater than 1
• Key insight: Scale factors less than 1 shrink figures, greater than 1 enlarge them

3. TRANSLATE the size comparison into an inequality

• Side of T = k × Side of S
• Given: Side of T > Side of S
• Therefore: k × Side of S > Side of S

4. SIMPLIFY to find the constraint on k

• Since side length is positive, divide both sides by Side of S:
• \(\mathrm{k \gt 1}\)

5. APPLY CONSTRAINTS to select the answer

• Check each choice against \(\mathrm{k \gt 1}\):

1. 29/28 ≈ 1.036 > 1 ✓
2. 1 = 1 (not greater than 1) ✗
3. 28/29 ≈ 0.966 < 1 ✗
4. 0 < 1 ✗

Answer: A. 29/28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret the problem and think that any scale factor will work, or confuse which triangle is larger.

Some students read "triangle T is created from triangle S" and assume T must be smaller, leading them to select Choice C (28/29) thinking the scale factor should be less than 1.

Second Most Common Error:

Missing conceptual knowledge about scale factors: Students may not understand that scale factors greater than 1 enlarge figures while factors less than 1 shrink them.

This conceptual gap causes confusion about which direction the inequality should go, leading to random guessing between the fractional choices.

The Bottom Line:

This problem tests whether students understand the fundamental relationship between scale factors and size changes. The key insight is recognizing that "larger" means the scale factor exceeds 1.