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
  • \(\mathrm{AC} = 21\) units
  • \(\mathrm{sin\ B} = \frac{3}{5}\)
  • Need to find perimeter

2. INFER the triangle relationships

• Since angle C is the right angle:
  • AB must be the hypotenuse (opposite the right angle)
  • AC and BC are the two legs
  • For angle B specifically: AC is opposite, BC is adjacent
• This means: \(\mathrm{sin\ B} = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}} = \frac{\mathrm{AC}}{\mathrm{AB}}\)

3. SIMPLIFY to find the hypotenuse

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

4. INFER the strategy for the missing side

• We now have: \(\mathrm{AB} = 35\), \(\mathrm{AC} = 21\), need BC
• Since this is a right triangle, use Pythagorean theorem: \(\mathrm{BC}^2 + \mathrm{AC}^2 = \mathrm{AB}^2\)

5. SIMPLIFY using Pythagorean theorem

• \(\mathrm{BC}^2 + 21^2 = 35^2\)
• \(\mathrm{BC}^2 + 441 = 1225\)
• \(\mathrm{BC}^2 = 784\)
• \(\mathrm{BC} = \sqrt{784} = 28\) (use calculator)

6. SIMPLIFY to find perimeter

• Perimeter = sum of all three sides
• \(\mathrm{Perimeter} = \mathrm{AB} + \mathrm{BC} + \mathrm{AC} = 35 + 28 + 21 = 84\)

Answer: C (84)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse which side is opposite to angle B, thinking BC (not AC) is opposite to B.

This leads them to set up \(\mathrm{sin\ B} = \frac{\mathrm{BC}}{\mathrm{AB}} = \frac{3}{5}\). While they might still correctly solve for \(\mathrm{AB} = 35\), they would then assign \(\mathrm{BC} = 21\) (the 3/5 proportion). However, when they try to verify with Pythagorean theorem: \(21^2 + \mathrm{AC}^2 = 35^2\), they get \(\mathrm{AC}^2 = 1225 - 441 = 784\), so \(\mathrm{AC} = 28\). But this contradicts the given information that \(\mathrm{AC} = 21\), creating confusion and likely causing them to make arithmetic errors or guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors during cross multiplication (getting AB = something other than 35) or when applying the Pythagorean theorem.

For example, if they miscalculate \(\mathrm{AB} = 25\) instead of 35, they would get: \(\mathrm{BC}^2 + 21^2 = 25^2\), so \(\mathrm{BC}^2 = 625 - 441 = 184\), giving \(\mathrm{BC} \approx 13.6\). Their perimeter would then be \(25 + 13.6 + 21 \approx 59.6\), which doesn't match any answer choice, leading to confusion and guessing.

The Bottom Line:

Success on this problem requires clear understanding of which side is opposite to which angle in a right triangle, combined with careful execution of both trigonometric ratios and the Pythagorean theorem.