A right triangle has legs with lengths of 11 centimeters and 9 centimeters. What is the length of this triangle's...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs of 11 cm and 9 cm
  • Need to find the hypotenuse length
• This tells us we need to use the Pythagorean theorem: \(\mathrm{c^2 = a^2 + b^2}\)

2. TRANSLATE the setup into the equation

• In the Pythagorean theorem \(\mathrm{c^2 = a^2 + b^2}\):
  • c represents the hypotenuse (what we're finding)
  • a and b represent the two legs (11 cm and 9 cm)
• Substituting: \(\mathrm{c^2 = 11^2 + 9^2}\)

3. SIMPLIFY the calculation

• Calculate each square:
  • \(\mathrm{11^2 = 121}\)
  • \(\mathrm{9^2 = 81}\)
• Add them: \(\mathrm{c^2 = 121 + 81 = 202}\)
• Take the square root: \(\mathrm{c = \sqrt{202}}\)

4. APPLY CONSTRAINTS to finalize the answer

• Since c represents a length, it must be positive
• Therefore: \(\mathrm{c = \sqrt{202}}\) cm

Answer: B. \(\mathrm{\sqrt{202}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse squaring numbers with doubling them, setting up \(\mathrm{c^2 = 11(2) + 9(2)}\) instead of \(\mathrm{c^2 = 11^2 + 9^2}\)

When they calculate: \(\mathrm{c^2 = 22 + 18 = 40}\), so \(\mathrm{c = \sqrt{40}}\)
This leads them to select Choice A (\(\mathrm{\sqrt{40}}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the equation incorrectly as \(\mathrm{c = 11^2 + 9^2}\) instead of \(\mathrm{c^2 = 11^2 + 9^2}\)

When they calculate: \(\mathrm{c = 121 + 81 = 202}\)
This leads them to select Choice D (202)

The Bottom Line:

The Pythagorean theorem requires precise setup - it's c² (c squared) equals the sum of the squares of the legs, not c equals the sum or any other variation. The mathematical notation must be translated correctly from the conceptual understanding.