An equilateral triangle has a height of 30 inches. What is the perimeter, in inches, of the triangle?20sqrt(3)30sqrt(3)60sqrt(3)90sqrt(3)

1. INFER the geometric relationships

• Given information: Equilateral triangle with height = 30 inches
• Key insight: In an equilateral triangle, the height from any vertex to the opposite side creates two congruent 30-60-90 right triangles
• The height becomes the longer leg of these special triangles

2. INFER which 30-60-90 ratio applies

• In a 30-60-90 triangle, sides are in ratio \(\mathrm{x : x\sqrt{3} : 2x}\)
• \(\mathrm{x}\) = short leg (opposite 30° angle)
• \(\mathrm{x\sqrt{3}}\) = long leg (opposite 60° angle) = our height
• \(\mathrm{2x}\) = hypotenuse = side of original equilateral triangle

3. TRANSLATE the height into the ratio

• We know: \(\mathrm{x\sqrt{3} = 30}\) inches
• Need to solve for \(\mathrm{x}\) to find the other measurements

4. SIMPLIFY to find x

• \(\mathrm{x\sqrt{3} = 30}\)
• \(\mathrm{x = \frac{30}{\sqrt{3}}}\)
• Rationalize the denominator: \(\mathrm{x = \frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{30\sqrt{3}}{3} = 10\sqrt{3}}\) inches

5. INFER the side length

• The side of the equilateral triangle = hypotenuse of 30-60-90 triangle
• Side length = \(\mathrm{2x = 2(10\sqrt{3}) = 20\sqrt{3}}\) inches

6. SIMPLIFY to find perimeter

• Perimeter = \(\mathrm{3 \times}\) side length = \(\mathrm{3 \times 20\sqrt{3} = 60\sqrt{3}}\) inches

Answer: C) \(\mathrm{60\sqrt{3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the height creates 30-60-90 triangles, so students attempt to use the Pythagorean theorem with unknown values or try other inappropriate methods. Without this key geometric insight, they can't establish the necessary relationships to solve the problem systematically. This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about 30-60-90 ratios: Students know about the special triangle but mix up which side corresponds to which part of the ratio. They might think the height is the short leg \(\mathrm{(x)}\) instead of the long leg \(\mathrm{(x\sqrt{3})}\), leading them to set up \(\mathrm{x = 30}\) instead of \(\mathrm{x\sqrt{3} = 30}\). This gives them \(\mathrm{x = 30}\), making the side length 60 and perimeter 180, which isn't among the choices. This may lead them to select Choice D (\(\mathrm{90\sqrt{3}}\)) by incorrectly applying radical operations.

The Bottom Line:

This problem requires both strong geometric reasoning to see the 30-60-90 relationship and careful attention to which part of the ratio corresponds to the given height. The key breakthrough is realizing that the height is the longer leg, not the shorter one.