An equilateral triangle has an altitude of length 24sqrt(3) inches. What is the perimeter, in inches, of this triangle? 72...

1. TRANSLATE the problem information

• Given information:
  • Equilateral triangle with \(\mathrm{altitude = 24\sqrt{3}\ inches}\)
  • Need to find the perimeter (sum of all three sides)

2. INFER the geometric relationship

• In an equilateral triangle, the altitude from any vertex to the opposite side creates two congruent 30-60-90 right triangles
• For an equilateral triangle with side length s, the altitude length = \(\frac{\mathrm{s\sqrt{3}}}{2}\)
• This is our key relationship to connect the given altitude with the unknown side length

3. SIMPLIFY by setting up and solving the equation

• Set altitude formula equal to given value: \(\frac{\mathrm{s\sqrt{3}}}{2} = 24\sqrt{3}\)
• Divide both sides by \(\sqrt{3}\): \(\frac{\mathrm{s}}{2} = 24\)
• Solve for s: \(\mathrm{s = 48\ inches}\)

4. TRANSLATE to find the final answer

• Perimeter = \(\mathrm{3s = 3(48) = 144\ inches}\)

Answer: D) 144

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing the altitude formula for equilateral triangles

Students may recognize this involves an equilateral triangle but not recall that altitude = \(\frac{\mathrm{s\sqrt{3}}}{2}\). Without this relationship, they cannot connect the given altitude to the side length. This leads to confusion and guessing among the answer choices.

Second Most Common Error Path:

Weak TRANSLATE reasoning: Finding the side length correctly but forgetting that perimeter means all three sides

Students successfully find \(\mathrm{s = 48}\), but then look for 48 among the answer choices. When they don't see it, they might select Choice B (48√3) thinking it's somehow related to their calculated value, or get confused about what the problem is actually asking.

The Bottom Line:

This problem requires both knowing the specific altitude formula for equilateral triangles AND remembering to apply the definition of perimeter. Many students struggle because they either lack the geometric relationship or make the final translation error.