In a right triangle, the lengths of the legs are x units and 5 units, and the hypotenuse is 13...

1. TRANSLATE the problem information

• Given information:
  • Right triangle
  • One leg = x units (unknown)
  • Other leg = 5 units
  • Hypotenuse = 13 units
• What we need to find: The value of x

2. INFER the approach

• This is a right triangle with two sides known and one unknown
• The Pythagorean theorem applies: \(\mathrm{a^2 + b^2 = c^2}\)
• We'll substitute the known values and solve for x

3. TRANSLATE the Pythagorean theorem setup

Set up the equation using legs = x and 5, hypotenuse = 13:
\(\mathrm{x^2 + 5^2 = 13^2}\)

4. SIMPLIFY to solve for x

• Calculate the squares: \(\mathrm{x^2 + 25 = 169}\)
• Subtract 25 from both sides: \(\mathrm{x^2 = 169 - 25 = 144}\)
• Take the square root: \(\mathrm{x = \sqrt{144} = 12}\)
• Since length must be positive: x = 12

Answer: (C) 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when solving \(\mathrm{x^2 = 144}\), particularly with square roots.

Some students might calculate √144 incorrectly, perhaps thinking it equals 10 or getting confused with other perfect squares. Others might make errors in the subtraction step (169 - 25), accidentally calculating it as 100 instead of 144.

This may lead them to select Choice (B) (10) if they incorrectly solve \(\mathrm{x^2 = 100}\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which measurement corresponds to which part of the triangle.

Some students might think the unknown x represents the hypotenuse rather than a leg, or they might set up the equation incorrectly. Others might assume both legs are equal (x = 5) without using the Pythagorean theorem at all.

This may lead them to select Choice (A) (5) by assuming both legs have the same length, or Choice (D) (13) by confusing the unknown with the hypotenuse.

The Bottom Line:

This problem tests whether students can correctly identify a Pythagorean theorem situation and execute the algebra accurately. The key challenge is maintaining precision through the multi-step algebraic solution while keeping track of which measurement represents which part of the triangle.