A right triangle has legs of lengths x and 24. The length of the hypotenuse is sqrt(x^2 + 24^2). What...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs of lengths x and 24
  • Hypotenuse formula: \(\sqrt{\mathrm{x}^2 + 24^2}\)
  • Need to find the length when x = 7

• What this tells us: We need to substitute x = 7 into the given expression

2. SIMPLIFY through substitution and calculation

• Substitute x = 7 into \(\sqrt{\mathrm{x}^2 + 24^2}\):

\(\sqrt{7^2 + 24^2}\)

• Calculate each squared term:

\(7^2 = 49\)

\(24^2 = 576\)

• Add under the radical:

\(\sqrt{49 + 576} = \sqrt{625}\)

• Take the square root:

\(\sqrt{625} = 25\)

Answer: B. 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors in the calculation steps.

Common mistakes include:

• Calculating \(7^2\) incorrectly (maybe getting 14 instead of 49)
• Making errors in \(24^2 = 576\)
• Adding \(49 + 576\) incorrectly
• Not recognizing that \(\sqrt{625} = 25\)

These calculation errors can lead to selecting any of the incorrect answer choices or getting stuck and guessing.

The Bottom Line:

This problem tests careful arithmetic execution more than conceptual understanding, since the Pythagorean relationship is already provided in the problem statement.