A triangle with angle measures 45°, 45°, and 90° has a perimeter of 8 + 8sqrt(2). What is the length...

1. TRANSLATE the problem information

• Given information:
  • Triangle with angles \(45°, 45°, 90°\)
  • Perimeter = \(8 + 8\sqrt{2}\)
• We need to find the hypotenuse length

2. INFER the triangle properties and setup

• In a 45-45-90 triangle, sides are in ratio \(1:1:\sqrt{2}\)
• If each leg = \(\mathrm{s}\), then hypotenuse = \(\mathrm{s}\sqrt{2}\)
• Perimeter = leg + leg + hypotenuse = \(\mathrm{s} + \mathrm{s} + \mathrm{s}\sqrt{2} = \mathrm{s}(2 + \sqrt{2})\)

3. TRANSLATE into an equation

Set perimeter formula equal to given value:

\(\mathrm{s}(2 + \sqrt{2}) = 8 + 8\sqrt{2}\)

4. INFER the solution strategy

• Factor the right side: \(8 + 8\sqrt{2} = 8(1 + \sqrt{2})\)
• So: \(\mathrm{s}(2 + \sqrt{2}) = 8(1 + \sqrt{2})\)
• Divide both sides by \((1 + \sqrt{2})\): \(\mathrm{s} \cdot \frac{(2 + \sqrt{2})}{(1 + \sqrt{2})} = 8\)

5. SIMPLIFY the fraction \(\frac{(2 + \sqrt{2})}{(1 + \sqrt{2})}\)

• Rationalize by multiplying by \(\frac{(1 - \sqrt{2})}{(1 - \sqrt{2})}\):
• Numerator: \((2 + \sqrt{2})(1 - \sqrt{2}) = 2 - 2\sqrt{2} + \sqrt{2} - 2 = -\sqrt{2}\)
• Denominator: \((1 + \sqrt{2})(1 - \sqrt{2}) = 1 - 2 = -1\)
• Result: \(\frac{(-\sqrt{2})}{(-1)} = \sqrt{2}\)

6. SIMPLIFY to find s and the hypotenuse

• \(\mathrm{s} \cdot \sqrt{2} = 8\), so \(\mathrm{s} = \frac{8}{\sqrt{2}} = 4\sqrt{2}\)
• Hypotenuse = \(\mathrm{s}\sqrt{2} = 4\sqrt{2} \cdot \sqrt{2} = 8\)

Answer: D (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students struggle with the rationalization step \(\frac{(2 + \sqrt{2})}{(1 + \sqrt{2})}\).

Many students get confused when multiplying \((2 + \sqrt{2})(1 - \sqrt{2})\) and make sign errors or forget terms, leading to incorrect values for s. They might incorrectly calculate the numerator as \(2 + \sqrt{2}\) instead of \(-\sqrt{2}\), or mess up the denominator calculation. This leads to wrong values for s and ultimately the wrong hypotenuse.

This may lead them to select Choice A (4) or Choice B \((4\sqrt{2})\) depending on their calculation errors.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that 45-45-90 triangles have sides in ratio \(1:1:\sqrt{2}\).

Without this key relationship, they can't set up the perimeter equation properly. Some might try to use the Pythagorean theorem with unknown sides, or assume all sides are equal, leading to completely incorrect approaches.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem combines special triangle knowledge with complex algebraic manipulation. Success requires both remembering the 45-45-90 triangle ratios AND executing a challenging rationalization correctly.