A right triangle has legs with lengths of 28 centimeters and 20 centimeters. What is the length of this triangle's...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs 28 cm and 20 cm
  • Need to find hypotenuse length

• This tells us we need the Pythagorean theorem since we have two legs and need the hypotenuse.

2. INFER the approach

• Since we have a right triangle with two known legs, the Pythagorean theorem is the direct path: \(\mathrm{a^2 + b^2 = c^2}\)
• We'll substitute our leg lengths, solve for c², then take the square root.

3. Apply the Pythagorean theorem

• Set up: \(\mathrm{28^2 + 20^2 = c^2}\)
• Calculate: \(\mathrm{784 + 400 = 1,184}\)
• So \(\mathrm{c^2 = 1,184}\)

4. SIMPLIFY to find the hypotenuse

• Take the square root: \(\mathrm{c = \sqrt{1,184}}\)
• Factor 1,184 to simplify: \(\mathrm{1,184 = 16 \times 74}\)
• Therefore: \(\mathrm{\sqrt{1,184} = \sqrt{16 \times 74} = \sqrt{16} \times \sqrt{74} = 4\sqrt{74}}\)

Answer: B. \(\mathrm{4\sqrt{74}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up and solve \(\mathrm{c^2 = 1,184}\) but stop there without taking the square root, thinking 1,184 is the final answer.

They see the large number 1,184 and assume it must be the hypotenuse length, forgetting that they solved for \(\mathrm{c^2}\), not \(\mathrm{c}\).

This may lead them to select Choice D (1,184).

Second Most Common Error:

Poor SIMPLIFY technique: Students take the square root to get \(\mathrm{\sqrt{1,184}}\) but don't know how to simplify it further, so they either guess or try to estimate it as a whole number.

Without factoring skills, they might estimate \(\mathrm{\sqrt{1,184} \approx 34}\) and look for the closest answer, potentially selecting Choice C (48) as it seems reasonable for a triangle with legs of 28 and 20.

The Bottom Line:

This problem tests both the fundamental application of the Pythagorean theorem and the essential skill of simplifying radicals. Many students can set up the equation correctly but struggle with the final step of radical simplification, which is crucial for matching the exact form of the answer choices.