The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is ksqrt(3) centimeters, where k is...

1. TRANSLATE the problem information

• Given information:
  • Equilateral triangle (all sides equal)
  • Perimeter = \(\mathrm{624\text{ cm}}\)
  • Height = \(\mathrm{k\sqrt{3}\text{ cm}}\)
• We need to find the constant k

2. INFER the approach needed

• To find k, I need to determine the actual height of the triangle
• This requires finding the side length first (from perimeter)
• Then use the geometric relationship between height and side length in equilateral triangles

3. TRANSLATE perimeter to side length

• Since all three sides are equal: Perimeter = \(\mathrm{3 \times side\text{ }length}\)
• \(\mathrm{3s = 624}\)
• \(\mathrm{s = 208\text{ cm}}\)

4. INFER the height relationship using geometry

• When we drop an altitude in an equilateral triangle, it creates two congruent right triangles
• Each right triangle has:
  • Hypotenuse = side length = \(\mathrm{208\text{ cm}}\)
  • One leg = half the base = \(\mathrm{208/2 = 104\text{ cm}}\)
  • Other leg = height (what we're finding)

5. SIMPLIFY using Pythagorean theorem

• \(\mathrm{h^2 + 104^2 = 208^2}\)
• \(\mathrm{h^2 + 10,816 = 43,264}\)
• \(\mathrm{h^2 = 32,448}\)
• \(\mathrm{h = \sqrt{32,448} = \sqrt{10,816 \times 3} = 104\sqrt{3}\text{ cm}}\)

6. SIMPLIFY to find k

• Given: height = \(\mathrm{k\sqrt{3}}\)
• From calculation: height = \(\mathrm{104\sqrt{3}}\)
• Therefore: \(\mathrm{k\sqrt{3} = 104\sqrt{3}}\)
• \(\mathrm{k = 104}\)

Answer: 104

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly assume that k represents the side length rather than understanding that \(\mathrm{k\sqrt{3}}\) represents the height.

Some students might set up: \(\mathrm{k = 208}\) (thinking k is the side length), leading to an incorrect answer that doesn't match any reasonable expectation for this type of problem. This leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about equilateral triangles: Students may not know how to find the height of an equilateral triangle given the side length.

Without knowing the height formula or geometric relationship, students cannot establish the connection between the \(\mathrm{208\text{ cm}}\) side length and the height expression \(\mathrm{k\sqrt{3}}\). This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can work backwards from a given perimeter to find a parameter in the height formula. Success requires both translating the perimeter correctly AND knowing the geometric relationship for equilateral triangle heights.