A right triangle has two legs of equal length. If the area of this triangle is 18 square inches, what...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with two legs of equal length
  • Area = 18 square inches
  • Need to find the length of each leg
• What this tells us: We have an isosceles right triangle where both legs have the same length

2. INFER the approach

• For any triangle: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• In a right triangle, the two legs are perpendicular, so they can serve as base and height
• Since both legs have equal length s: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{s} \times \mathrm{s} = \frac{1}{2}\mathrm{s}^2\)

3. TRANSLATE the given area into an equation

• We know the area is 18 square inches:
  \(18 = \frac{1}{2}\mathrm{s}^2\)

4. SIMPLIFY to solve for s

• Multiply both sides by 2: \(36 = \mathrm{s}^2\)
• Take the square root of both sides: \(\mathrm{s} = \sqrt{36} = 6\)

Answer: 6 inches (Choice A)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that the legs of a right triangle serve as base and height in the area formula. Instead, they might try to use more complex approaches involving the hypotenuse or get confused about which measurements to use.

This confusion leads them to set up incorrect equations or abandon systematic solution, causing them to guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(18 = \frac{1}{2}\mathrm{s}^2\) but make algebraic errors. Common mistakes include:

• Forgetting to multiply by 2, leading to \(\mathrm{s}^2 = 18\), so \(\mathrm{s} = \sqrt{18} \approx 4.24\) (not matching any choice)
• Getting \(\mathrm{s}^2 = 36\) but forgetting to take the square root, thinking the answer is 36

This may lead them to select Choice D (36) by stopping too early in their calculation.

The Bottom Line:

This problem tests whether students can connect the abstract area formula to the concrete setup of a right triangle, then execute the algebra correctly. The key insight is recognizing that equal legs make this a clean algebraic problem rather than a complex geometry problem.