The perimeter of an isosceles right triangle is 18 + 18sqrt(2) inches. What is the length, in inches, of the...

1. TRANSLATE the problem information

• Given information:
  • Isosceles right triangle (two equal legs, \(90°\) angle between them)
  • Perimeter = \(18 + 18\sqrt{2}\) inches
  • Need to find: hypotenuse length

• What this tells us: We need to set up a relationship between legs and hypotenuse

2. INFER the geometric relationships

• In an isosceles right triangle, if each leg has length \(\mathrm{x}\), then:
  • Hypotenuse = \(\mathrm{x\sqrt{2}}\) (from the 45-45-90 triangle relationship)
  • Perimeter = leg + leg + hypotenuse = \(\mathrm{x + x + x\sqrt{2} = x(2 + \sqrt{2})}\)

3. TRANSLATE into an equation

• Set up the perimeter equation:

\(\mathrm{x(2 + \sqrt{2}) = 18 + 18\sqrt{2}}\)

• Factor the right side: \(\mathrm{18 + 18\sqrt{2} = 18(1 + \sqrt{2})}\)

4. SIMPLIFY to solve for x

• We have: \(\mathrm{x(2 + \sqrt{2}) = 18(1 + \sqrt{2})}\)
• So: \(\mathrm{x = \frac{18(1 + \sqrt{2})}{(2 + \sqrt{2})}}\)

• Rationalize by multiplying by \(\mathrm{\frac{(2 - \sqrt{2})}{(2 - \sqrt{2})}}\):

\(\mathrm{x = \frac{18(1 + \sqrt{2})(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})}}\)

• Denominator: \(\mathrm{(2 + \sqrt{2})(2 - \sqrt{2}) = 4 - 2 = 2}\)
• Numerator: \(\mathrm{18(1 + \sqrt{2})(2 - \sqrt{2}) = 18[2 - \sqrt{2} + 2\sqrt{2} - 2] = 18\sqrt{2}}\)
• Therefore: \(\mathrm{x = \frac{18\sqrt{2}}{2} = 9\sqrt{2}}\)

5. INFER the final answer

• Each leg has length \(\mathrm{9\sqrt{2}}\)
• Hypotenuse = \(\mathrm{x\sqrt{2} = (9\sqrt{2})(\sqrt{2}) = 9 \times 2 = 18}\)

Answer: C. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find \(\mathrm{x = 9\sqrt{2}}\) correctly but then select this as their final answer, forgetting that the question asks for the hypotenuse, not the leg length.

They think: "I solved for x and got \(\mathrm{9\sqrt{2}}\), so that must be the answer." This may lead them to select Choice B (\(\mathrm{9\sqrt{2}}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly but make errors when rationalizing the denominator or expanding \(\mathrm{(1 + \sqrt{2})(2 - \sqrt{2})}\).

Common mistakes include forgetting to distribute properly or arithmetic errors like thinking \(\mathrm{\sqrt{2} \times \sqrt{2} = \sqrt{4}}\) instead of 2. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires both strong algebraic manipulation skills with radicals AND remembering to answer the actual question being asked. Many students get mathematically correct intermediate results but lose points by not completing the final step.