A right triangle has interior angles measuring 30°, 60°, and 90°. The perimeter of this triangle is 75 + 75sqrt(3)...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with angles \(30°\), \(60°\), and \(90°\)
  • Perimeter = \(75 + 75\sqrt{3}\) centimeters
  • Need to find the shortest leg
• What this tells us: We have a special 30-60-90 triangle where we can use known side ratios

2. INFER the approach using special triangle properties

• In a 30-60-90 triangle, sides are always in ratio \(1:\sqrt{3}:2\)
• If shortest leg = s, then:
  • Shortest leg (opposite \(30°\)): \(\mathrm{s}\)
  • Longer leg (opposite \(60°\)): \(\mathrm{s}\sqrt{3}\)
  • Hypotenuse (opposite \(90°\)): \(2\mathrm{s}\)
• Strategy: Set up perimeter equation and solve for s

3. TRANSLATE perimeter into equation

• Perimeter = \(\mathrm{s} + \mathrm{s}\sqrt{3} + 2\mathrm{s} = \mathrm{s}(1 + \sqrt{3} + 2) = \mathrm{s}(3 + \sqrt{3})\)
• Given perimeter = \(75 + 75\sqrt{3} = 75(1 + \sqrt{3})\)
• Equation: \(\mathrm{s}(3 + \sqrt{3}) = 75(1 + \sqrt{3})\)

4. SIMPLIFY to solve for s

• Divide both sides by \((3 + \sqrt{3})\):
  \(\mathrm{s} = \frac{75(1 + \sqrt{3})}{(3 + \sqrt{3})}\)

• To rationalize, multiply by \(\frac{(3 - \sqrt{3})}{(3 - \sqrt{3})}\):
  \(\mathrm{s} = \frac{75(1 + \sqrt{3})(3 - \sqrt{3})}{[(3 + \sqrt{3})(3 - \sqrt{3})]}\)

• SIMPLIFY the denominator:
  \((3 + \sqrt{3})(3 - \sqrt{3}) = 9 - 3 = 6\)

• SIMPLIFY the numerator:
  \(75(1 + \sqrt{3})(3 - \sqrt{3}) = 75[3 - \sqrt{3} + 3\sqrt{3} - 3] = 75(2\sqrt{3}) = 150\sqrt{3}\)

• Therefore:
  \(\mathrm{s} = \frac{150\sqrt{3}}{6} = 25\sqrt{3}\)

Answer: B. \(25\sqrt{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not immediately recognize this as a special 30-60-90 triangle requiring the \(1:\sqrt{3}:2\) ratio. Instead, they might try to use trigonometry (like sin, cos, tan) or attempt to solve it as a general triangle, leading to unnecessary complexity and likely errors.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{s}(3 + \sqrt{3}) = 75(1 + \sqrt{3})\) but make algebraic errors when solving for s. They might incorrectly cancel terms or make mistakes with radical expressions, potentially getting \(\mathrm{s} = 75\) or \(\mathrm{s} = 25\).

This may lead them to select Choice A (25) or Choice C (75).

The Bottom Line:

This problem requires recognizing the special triangle pattern immediately and being comfortable manipulating expressions with square roots. Students who don't see the \(1:\sqrt{3}:2\) ratio connection will struggle to even begin systematically.