One leg of a right triangle has a length of 43.2text{ millimeters}. The hypotenuse of the triangle has a length...

1. TRANSLATE the problem information

• Given information:
  • One leg = 43.2 millimeters
  • Hypotenuse = 196.8 millimeters
  • Find: the other leg

2. INFER the approach

• This is a right triangle problem, so we need the Pythagorean theorem
• We have one leg and the hypotenuse, need to find the other leg
• Set up: \(\mathrm{a^2 + b^2 = c^2}\) where c is the hypotenuse

3. SIMPLIFY through the algebraic steps

• Substitute known values: \(\mathrm{(43.2)^2 + b^2 = (196.8)^2}\)
• Calculate the squares (use calculator):
  • \(\mathrm{(43.2)^2 = 1,866.24}\)
  • \(\mathrm{(196.8)^2 = 38,730.24}\)
• The equation becomes: \(\mathrm{1,866.24 + b^2 = 38,730.24}\)
• Subtract to isolate b²: \(\mathrm{b^2 = 38,730.24 - 1,866.24 = 36,864}\)
• Take the square root (use calculator): \(\mathrm{b = \sqrt{36,864} = 192}\)

Answer: C. 192

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the squares or the final square root.

Many students correctly set up \(\mathrm{(43.2)^2 + b^2 = (196.8)^2}\) but then make calculation mistakes. They might get \(\mathrm{1,866.24 + b^2 = 38,730.24}\) correct, but then miscalculate 38,730.24 - 1,866.24 or make an error when finding \(\mathrm{\sqrt{36,864}}\). These arithmetic slips can lead them to select any of the incorrect answer choices or cause confusion that leads to guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which measurement is which in the Pythagorean theorem setup.

Some students might incorrectly think the 196.8 mm measurement is a leg rather than the hypotenuse, leading them to set up the equation as \(\mathrm{(43.2)^2 + (196.8)^2 = c^2}\) instead. This fundamental misunderstanding of the problem setup leads to completely wrong calculations and typically results in confusion and random answer selection.

The Bottom Line:

This problem requires careful attention to both setup and calculation accuracy. Success depends on correctly identifying which measurements go where in the Pythagorean theorem, then executing multiple calculation steps without arithmetic errors.