A right triangle has legs with lengths of 24 centimeters and 21 centimeters. If the length of this triangle's hypotenuse,...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs 24 cm and 21 cm
  • Need hypotenuse in form 3√d
  • Find the value of d

2. INFER the approach

• For any right triangle, use Pythagorean theorem: \(\mathrm{a^2 + b^2 = c^2}\)
• Since we need the answer in form \(\mathrm{3\sqrt{d}}\), we'll need to factor out perfect squares from under the radical

3. SIMPLIFY using the Pythagorean theorem

• Set up the equation: \(\mathrm{h^2 = 24^2 + 21^2}\)
• Calculate: \(\mathrm{h^2 = 576 + 441 = 1{,}017}\) (use calculator)
• Therefore: \(\mathrm{h = \sqrt{1{,}017}}\)

4. INFER the factorization strategy

• To write \(\mathrm{\sqrt{1{,}017}}\) as \(\mathrm{3\sqrt{d}}\), we need \(\mathrm{1{,}017 = 9 \times d}\) (since \(\mathrm{3^2 = 9}\))
• This means we need to factor 1,017 and look for the factor 9

5. SIMPLIFY the radical expression

• Factor 1,017: Try dividing by 9
• \(\mathrm{1{,}017 \div 9 = 113}\) (use calculator to verify: \(\mathrm{9 \times 113 = 1{,}017}\))
• So \(\mathrm{1{,}017 = 9 \times 113}\)

6. SIMPLIFY using square root properties

• \(\mathrm{h = \sqrt{1{,}017}}\)
• \(\mathrm{h = \sqrt{9 \times 113}}\)
• \(\mathrm{h = \sqrt{9} \times \sqrt{113}}\)
• \(\mathrm{h = 3\sqrt{113}}\)
• Comparing to the form \(\mathrm{3\sqrt{d}}\): \(\mathrm{d = 113}\)

Answer: 113

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that 1,017 needs to be factored to extract perfect square factors

Students often get \(\mathrm{h = \sqrt{1{,}017}}\) and then don't know what to do next. They might try to approximate \(\mathrm{\sqrt{1{,}017} \approx 31.9}\) instead of recognizing they need to factor it. This leads to confusion and guessing rather than systematic problem-solving.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors in the initial Pythagorean calculation

Students might incorrectly calculate \(\mathrm{24^2 + 21^2}\) due to mental math errors (like getting \(\mathrm{24^2 = 586}\) instead of \(\mathrm{576}\)), leading to the wrong value under the radical and subsequently the wrong value for d.

The Bottom Line:

This problem tests whether students can bridge the gap between applying a familiar formula (Pythagorean theorem) and manipulating the result into a specific required form. The key insight is recognizing that radical expressions often need factoring to reveal their simplest form.