Question:The perimeter of a right triangle with angle measures 30°, 60°, text{ and } 90° is 12 + 4sqrt(3) feet....

1. TRANSLATE the problem information

• Given information:
  • Right triangle with angles \(30°\), \(60°\), and \(90°\)
  • Perimeter = \(12 + 4\sqrt{3}\) feet
  • Need to find: length of hypotenuse

2. INFER the triangle properties

• This is a 30-60-90 triangle, which has special side ratios
• If the shortest side (opposite \(30°\)) = \(\mathrm{x}\), then:
  • Side opposite \(60°\) = \(\mathrm{x\sqrt{3}}\)
  • Hypotenuse (opposite \(90°\)) = \(\mathrm{2x}\)

3. TRANSLATE the perimeter condition

• Perimeter = sum of all three sides
• \(\mathrm{x + x\sqrt{3} + 2x = 12 + 4\sqrt{3}}\)

4. SIMPLIFY the equation

• Combine like terms: \(\mathrm{x(1 + \sqrt{3} + 2) = 12 + 4\sqrt{3}}\)
• Simplify: \(\mathrm{x(3 + \sqrt{3}) = 12 + 4\sqrt{3}}\)
• Factor right side: \(\mathrm{x(3 + \sqrt{3}) = 4(3 + \sqrt{3})}\)
• Divide both sides by \(\mathrm{(3 + \sqrt{3})}\): \(\mathrm{x = 4}\)

5. INFER the final answer

• Since hypotenuse = \(\mathrm{2x}\) and \(\mathrm{x = 4}\)
• Hypotenuse = \(\mathrm{2(4) = 8}\) feet

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the special 30-60-90 triangle ratios and trying to use general right triangle methods instead.

Students might attempt to use the Pythagorean theorem without knowing the side relationships, leading them to set up complicated equations they can't solve. Without the key insight about \(\mathrm{1 : \sqrt{3} : 2}\) ratios, they get stuck early and resort to guessing.

This leads to confusion and random answer selection.

Second Most Common Error:

Poor SIMPLIFY execution: Setting up the correct equation \(\mathrm{x(3 + \sqrt{3}) = 12 + 4\sqrt{3}}\) but failing to factor the right side properly.

Students might try to divide by \(\mathrm{(3 + \sqrt{3})}\) without first recognizing that \(\mathrm{12 + 4\sqrt{3} = 4(3 + \sqrt{3})}\), leading to messy algebra with radicals in denominators. They may incorrectly simplify and get \(\mathrm{x = 6}\) instead of \(\mathrm{x = 4}\).

This may lead them to select Choice A (6) for the hypotenuse, forgetting that the hypotenuse is \(\mathrm{2x}\).

The Bottom Line:

This problem requires recognizing the special properties of 30-60-90 triangles. Without this key insight, students face unnecessarily complex algebra that usually leads to errors or abandoning the systematic approach entirely.