A right triangle has an area of 18 square units. If the length of one of its legs is 3sqrt(2)...

1. TRANSLATE the problem information

• Given information:
  • Right triangle area = 18 square units
  • One leg = \(3\sqrt{2}\) units
  • Find: hypotenuse length

2. INFER the solution strategy

• To find the hypotenuse, we need both legs first
• We have one leg, so use the area formula to find the other leg
• Then apply Pythagorean theorem for the hypotenuse

3. SIMPLIFY to find the unknown leg

• Area formula: \(\mathrm{A} = \frac{1}{2}\mathrm{ab}\) where \(\mathrm{a}\) and \(\mathrm{b}\) are the legs
• Substitute known values: \(18 = \frac{1}{2}(3\sqrt{2})(\mathrm{b})\)
• Solve for \(\mathrm{b}\):
  \(36 = 3\sqrt{2} \times \mathrm{b}\)
  so \(\mathrm{b} = \frac{36}{3\sqrt{2}} = \frac{12}{\sqrt{2}}\)
• Rationalize:
  \(\mathrm{b} = \frac{12}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}\)

4. SIMPLIFY using Pythagorean theorem

• With legs \(\mathrm{a} = 3\sqrt{2}\) and \(\mathrm{b} = 6\sqrt{2}\):
• \(\mathrm{c}^2 = (3\sqrt{2})^2 + (6\sqrt{2})^2\)
• \(\mathrm{c}^2 = 9(2) + 36(2) = 18 + 72 = 90\)
• \(\mathrm{c} = \sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}\)

Answer: \(3\sqrt{10}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when working with radicals, particularly in computing \((3\sqrt{2})^2 + (6\sqrt{2})^2\).

Many students incorrectly calculate \((3\sqrt{2})^2\) as \(9\sqrt{2}\) instead of 18, or make similar errors with \((6\sqrt{2})^2\). This leads to an incorrect value for \(\mathrm{c}^2\), resulting in the wrong hypotenuse length. They might end up selecting Choice (B) \((6\sqrt{2})\) by confusing one of the legs with the hypotenuse.

Second Most Common Error:

Poor INFER reasoning: Students attempt to use the Pythagorean theorem immediately without first finding the second leg.

Without recognizing that they need both legs before applying \(\mathrm{a}^2 + \mathrm{b}^2 = \mathrm{c}^2\), students get stuck trying to work backwards from incomplete information. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires systematic two-step reasoning combined with careful radical arithmetic - students must resist the urge to jump directly to the Pythagorean theorem and instead methodically work through finding the missing leg first.