Two identical triangular light fixtures are mounted at the same distance from a wall, each projecting a triangular beam of...

1. TRANSLATE the problem information

• Given information:
  • Two identical triangular light fixtures at the same distance from wall
  • First fixture: opening width = 8 feet, projected pattern width = 12 feet
  • Second fixture: opening width = unknown, projected pattern width = 9 feet

• What this tells us: We need to find the unknown opening width of the second fixture

2. INFER the key relationship

• Since the fixtures are identical and at the same distance from the wall, they create similar triangular light patterns
• This means the ratio of opening width to projected width must be the same for both fixtures
• We can set up a proportion to solve this

3. SIMPLIFY by setting up the proportion

• First fixture ratio: \(\frac{\mathrm{opening~width}}{\mathrm{projected~width}} = \frac{8}{12} = \frac{2}{3}\)
• Second fixture must have the same ratio: \(\frac{x}{9} = \frac{2}{3}\)

4. SIMPLIFY by solving the proportion

• Cross multiply: \(3x = 2(9) = 18\)
• Divide both sides by 3: \(x = 6\)

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that identical fixtures create similar triangular patterns with constant ratios. Instead, they might try to subtract or add the differences between given measurements without establishing the proportional relationship.

For example, they might notice that \(12 - 8 = 4\), and then subtract 4 from 9 to get 5, leading them to select Choice A (5).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the proportion but make arithmetic errors during cross multiplication or solving. They might incorrectly calculate \(2 \times 9 = 16\) instead of 18, or make division errors, leading to wrong answers.

This computational error could lead them to select Choice C (8) or cause confusion and guessing.

The Bottom Line:

The key insight is recognizing that this is fundamentally a similar triangles problem disguised as a real-world lighting scenario. Students must connect the physical setup (identical fixtures at same distance) to the mathematical concept (constant ratios in similar figures).