Question:In triangle ABC, angle B is a right angle, the measure of angle A is 28°, and the length of...

1. TRANSLATE the problem information

• Given information:
  • Right triangle ABC with right angle at B
  • Angle \(\mathrm{A = 28°}\)
  • \(\mathrm{BC = 20}\) units
  • Area = \(\mathrm{m\ cot\ 28°}\) (need to find m)

2. INFER the geometric relationships

• In this right triangle setup:
  • BC (length 20) is the side opposite to angle A
  • AB is the side adjacent to angle A
• To find the area, we need both legs of the triangle
• The cotangent ratio will help us find the missing leg AB

3. APPLY cotangent ratio

• \(\mathrm{cot\ 28° = \frac{adjacent}{opposite} = \frac{AB}{BC} = \frac{AB}{20}}\)
• Therefore: \(\mathrm{AB = 20\ cot\ 28°}\)

4. CALCULATE the triangle area

• \(\mathrm{Area = \frac{1}{2} \times base \times height = \frac{1}{2} \times AB \times BC}\)
• Substituting: \(\mathrm{Area = \frac{1}{2} \times (20\ cot\ 28°) \times 20}\)

5. SIMPLIFY to match the given form

• \(\mathrm{Area = \frac{1}{2} \times 400 \times cot\ 28°}\)
• \(\mathrm{Area = 200\ cot\ 28°}\)
• Comparing with \(\mathrm{m\ cot\ 28°}\), we get \(\mathrm{m = 200}\)

Answer: 200

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse which side is adjacent versus opposite to the given angle, leading them to use \(\mathrm{tan\ 28°}\) instead of \(\mathrm{cot\ 28°}\).

If they write \(\mathrm{tan\ 28° = \frac{BC}{AB} = \frac{20}{AB}}\), they get \(\mathrm{AB = \frac{20}{tan\ 28°}}\). Since \(\mathrm{\frac{1}{tan} = cot}\), this actually gives the same result, but students often don't recognize this relationship and may calculate incorrectly or get confused about the form.

This confusion can lead them to abandon systematic solution and guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the cotangent relationship and area formula but make algebraic errors when combining terms.

For example, they might write \(\mathrm{Area = \frac{1}{2} \times 20 \times 20 \times cot\ 28° = 20\ cot\ 28°}\) instead of \(\mathrm{200\ cot\ 28°}\), forgetting to properly multiply the coefficients.

This leads them to incorrectly conclude \(\mathrm{m = 20}\).

The Bottom Line:

This problem requires students to seamlessly connect trigonometric ratios with geometric area formulas. The challenge lies in maintaining accuracy through multiple algebraic steps while keeping track of which trigonometric function correctly relates the known and unknown sides.