In triangle ABC, angle C is a right angle and the length of AC is 21 units. If sin B...

1. TRANSLATE the problem information

• Given information:
  • Right triangle ABC with right angle at C
  • AC = 21 units
  • \(\mathrm{sin\ B = \frac{3}{5}}\)
• What this tells us: We need to find all three sides to calculate the perimeter

2. INFER the trigonometric relationship

• In a right triangle, \(\mathrm{sin\ B = \frac{opposite\ side}{hypotenuse}}\)
• From angle B's perspective: AC is the opposite side, AB is the hypotenuse
• This gives us: \(\mathrm{sin\ B = \frac{AC}{AB}}\)

3. TRANSLATE into an equation and SIMPLIFY

• Set up the equation: \(\mathrm{\frac{AC}{AB} = sin\ B}\)
• Substitute known values: \(\mathrm{\frac{21}{AB} = \frac{3}{5}}\)
• Cross multiply: \(\mathrm{21 \times 5 = 3 \times AB}\)
• SIMPLIFY: \(\mathrm{105 = 3 \times AB}\), so \(\mathrm{AB = 35}\)

4. INFER what's needed next and SIMPLIFY

• We have AC = 21 and AB = 35, but still need BC for the perimeter
• Use Pythagorean theorem: \(\mathrm{AC^2 + BC^2 = AB^2}\)
• Substitute: \(\mathrm{21^2 + BC^2 = 35^2}\)
• SIMPLIFY: \(\mathrm{441 + BC^2 = 1225}\)
• \(\mathrm{BC^2 = 784}\), so \(\mathrm{BC = 28}\)

5. Calculate the perimeter

• \(\mathrm{Perimeter = AB + BC + AC = 35 + 28 + 21 = 84}\)

Answer: C (84)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might confuse which side is opposite to angle B, incorrectly thinking that BC (rather than AC) is opposite to angle B. This leads them to set up \(\mathrm{sin\ B = \frac{BC}{AB}}\) instead of \(\mathrm{sin\ B = \frac{AC}{AB}}\).

With this incorrect setup, they would get \(\mathrm{\frac{BC}{AB} = \frac{3}{5}}\), and if AB = 35 (which they might guess or derive incorrectly), they'd calculate BC = 21. Then using Pythagorean theorem with BC = 21 and AB = 35, they'd get AC = 28, leading to a perimeter of 35 + 21 + 28 = 84. Interestingly, this still gives the correct answer due to the symmetric nature of this particular problem, but the reasoning is flawed.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{sin\ B = \frac{AC}{AB} = \frac{21}{AB} = \frac{3}{5}}\) but make arithmetic errors during cross multiplication or when applying the Pythagorean theorem. For example, they might calculate AB = 25 instead of 35, or make errors when computing \(\mathrm{BC = \sqrt{784} = 28}\).

This leads to incorrect side lengths and consequently wrong perimeter calculations that don't match any of the given choices, causing them to guess.

The Bottom Line:

This problem requires careful attention to the trigonometric setup (which side is opposite to which angle) and accurate multi-step arithmetic. The key insight is recognizing that finding one unknown side through trigonometry enables finding the remaining side through the Pythagorean theorem.