Triangle F and Triangle G are similar triangles. The area of Triangle F is 169 times the area of Triangle...

1. TRANSLATE the problem information

• Given information:
  • Triangle F and Triangle G are similar triangles
  • Area of Triangle F is 169 times the area of Triangle G → \(\mathrm{A_F/A_G = 169}\)
  • Shortest side of Triangle G is 15
• Find: shortest side of Triangle F

2. INFER the key relationship

• For similar figures, areas and sides scale differently
• If corresponding sides have ratio k, then areas have ratio k²
• Since we know the area ratio (169), we can work backwards to find the side ratio

3. SIMPLIFY to find the side ratio

• Set up the relationship: \(\mathrm{(side\ ratio)^2 = area\ ratio}\)
• \(\mathrm{(s_F/s_G)^2 = 169}\)
• Take the square root of both sides: \(\mathrm{s_F/s_G = \sqrt{169} = 13}\)
• This means Triangle F's sides are 13 times longer than Triangle G's corresponding sides

4. SIMPLIFY to find the shortest side of Triangle F

• Since \(\mathrm{s_G = 15}\) and the side ratio is 13:
• \(\mathrm{s_F = 13 \times 15 = 195}\)

Answer: C) 195

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the area scaling rule to finding side lengths. They might try to directly use the area ratio as the side ratio, thinking that if areas differ by 169, then sides also differ by 169.

This leads them to calculate: \(\mathrm{s_F = 169 \times 15 = 2,535}\), causing them to select Choice D (2,535).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the square root of 169, but make an arithmetic error. They might incorrectly calculate \(\mathrm{\sqrt{169} = 12}\) instead of 13.

This leads to: \(\mathrm{s_F = 12 \times 15 = 180}\), causing them to select Choice B (180).

The Bottom Line:

The key insight is recognizing that area scaling (\(\mathrm{k^2}\)) and side scaling (\(\mathrm{k}\)) are different for similar figures. Students who miss this relationship either use the area ratio directly or struggle with the square root step, leading to systematic errors rather than random guessing.