A right triangle has a hypotenuse of 15 centimeters and one leg of 6 centimeters. If the length of the...

1. TRANSLATE the problem information

• Given information:
  • Right triangle with hypotenuse = \(\mathrm{15~cm}\)
  • One leg = \(\mathrm{6~cm}\)
  • Other leg can be written as \(\mathrm{3\sqrt{d}}\)
• What we need: Find the value of d

2. INFER the mathematical approach

• This is a right triangle problem, so we need the Pythagorean theorem
• The hypotenuse (15) is the longest side, and we have one leg (6)
• Strategy: Use \(\mathrm{a^2 + b^2 = c^2}\) to find the unknown leg, then match it to \(\mathrm{3\sqrt{d}}\)

3. SIMPLIFY using the Pythagorean theorem

• Set up the equation: \(\mathrm{6^2 + b^2 = 15^2}\)
• Calculate: \(\mathrm{36 + b^2 = 225}\)
• Solve for \(\mathrm{b^2}\): \(\mathrm{b^2 = 225 - 36 = 189}\)
• Therefore: \(\mathrm{b = \sqrt{189}}\)

4. SIMPLIFY the radical expression

• Factor 189 to find perfect square factors
• \(\mathrm{189 = 9 \times 21 = 3^2 \times 21}\)
• Apply radical property: \(\mathrm{\sqrt{189} = \sqrt{9 \times 21} = \sqrt{9} \times \sqrt{21} = 3\sqrt{21}}\)
• Comparing \(\mathrm{3\sqrt{21}}\) with \(\mathrm{3\sqrt{d}}\): \(\mathrm{d = 21}\)

Answer: B) 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly find \(\mathrm{b^2 = 189}\) but struggle to factor 189 properly or don't recognize that they need to extract perfect square factors from under the radical.

They might leave the answer as \(\mathrm{\sqrt{189}}\) without simplifying, or incorrectly factor 189 (perhaps as \(\mathrm{3 \times 63}\) instead of \(\mathrm{9 \times 21}\)), leading them to select a wrong answer choice or become confused and guess.

Second Most Common Error:

Poor INFER reasoning: Students incorrectly identify which side is the hypotenuse, perhaps setting up the equation as \(\mathrm{15^2 + 6^2 = b^2}\) instead of \(\mathrm{6^2 + b^2 = 15^2}\).

This gives them \(\mathrm{b^2 = 225 + 36 = 261}\), leading to \(\mathrm{b = \sqrt{261}}\), which doesn't match any of the given answer choices in the form \(\mathrm{3\sqrt{d}}\). This causes them to get stuck and guess.

The Bottom Line:

This problem tests both your understanding of the Pythagorean theorem and your ability to manipulate radical expressions. The key insight is recognizing that you need to factor the number under the radical to extract perfect squares.