A triangle with angle measures 30°, 60°, and 90° has a perimeter of 18 + 6sqrt(3). What is the length...

1. TRANSLATE the problem information

• Given information:
  • Triangle has angles \(30°, 60°, \text{ and } 90°\)
  • Perimeter = \(18 + 6\sqrt{3}\)
  • Need to find the longest side

2. INFER the triangle type and approach

• This is a 30-60-90 special right triangle
• These triangles have sides in the ratio \(1 : \sqrt{3} : 2\)
• If we call the shortest side x, then:
  • Side opposite \(30° = x\) (shortest)
  • Side opposite \(60° = x\sqrt{3}\) (medium)
  • Side opposite \(90° = 2x\) (longest - this is what we want)

3. TRANSLATE the perimeter condition into an equation

• Perimeter = sum of all three sides
• \(x + x\sqrt{3} + 2x = 18 + 6\sqrt{3}\)

4. SIMPLIFY the left side of the equation

• Factor out x: \(x(1 + \sqrt{3} + 2) = 18 + 6\sqrt{3}\)
• Combine terms: \(x(3 + \sqrt{3}) = 18 + 6\sqrt{3}\)

5. INFER how to handle the right side

• Factor out 6 from the right side: \(18 + 6\sqrt{3} = 6(3 + \sqrt{3})\)
• Now we have: \(x(3 + \sqrt{3}) = 6(3 + \sqrt{3})\)

6. SIMPLIFY to solve for x

• Divide both sides by \((3 + \sqrt{3})\): \(x = 6\)

7. INFER the final answer

• The longest side is the hypotenuse = \(2x = 2(6) = 12\)

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing this as a 30-60-90 special triangle

Students might try to use the Pythagorean theorem or other general triangle methods instead of the special ratio. Without knowing the \(1:\sqrt{3}:2\) relationship, they get stuck trying to set up equations with unknown sides and can't make progress with just the perimeter information. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic mistakes when factoring or solving

Students correctly set up \(x(3 + \sqrt{3}) = 18 + 6\sqrt{3}\) but fail to recognize they can factor 6 from the right side. They might try to divide by \((3 + \sqrt{3})\) incorrectly or make calculation errors, leading to wrong values like \(x = 3\) or \(x = 9\), which would give longest sides of 6 or 18 respectively.

The Bottom Line:

This problem tests whether students can recognize special triangle patterns and apply them systematically. The key insight is that knowing the angle measures immediately tells you the side ratios, turning what seems like an impossible problem into a straightforward algebraic solution.