In right triangle ABC, the measure of angle C = 90°. The length of the hypotenuse AB = 52 units....

1. TRANSLATE the problem information

• Given information:
  • Right triangle ABC with right angle at C
  • Hypotenuse \(\mathrm{AB = 52}\) units
  • \(\mathrm{cos\,A = \frac{5}{13}}\)
  • Need to find area
• What this tells us: We have angle A's cosine ratio and the hypotenuse length, which should let us find the adjacent side to angle A.

2. INFER the approach

• Since we know cos A and the hypotenuse, we can find the side adjacent to angle A
• Once we have two sides, we can use the Pythagorean theorem to find the third side
• With both legs known, we can calculate the area

3. TRANSLATE the cosine definition into an equation

• In right triangle ABC: \(\mathrm{cos\,A = \frac{adjacent\,side}{hypotenuse} = \frac{AC}{AB}}\)
• Substitute known values: \(\mathrm{\frac{5}{13} = \frac{AC}{52}}\)

4. SIMPLIFY to find the adjacent side

• Solve for AC: \(\mathrm{AC = 52 \times \frac{5}{13}}\)
  \(\mathrm{= \frac{260}{13}}\)
  \(\mathrm{= 20}\) units

5. INFER that we need the other leg for area calculation

• We have \(\mathrm{AC = 20}\) and \(\mathrm{AB = 52}\)
• Use Pythagorean theorem: \(\mathrm{AC^2 + BC^2 = AB^2}\)

6. SIMPLIFY to find the opposite side

• \(\mathrm{20^2 + BC^2 = 52^2}\)
• \(\mathrm{400 + BC^2 = 2704}\)
• \(\mathrm{BC^2 = 2304}\)
• \(\mathrm{BC = \sqrt{2304} = 48}\) units (use calculator if needed)

7. SIMPLIFY to calculate the area

• \(\mathrm{Area = \frac{1}{2} \times base \times height = \frac{1}{2} \times AC \times BC}\)
• \(\mathrm{Area = \frac{1}{2} \times 20 \times 48}\)
  \(\mathrm{= 480}\) square units

Answer: C (480)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse which side is adjacent to angle A versus which side is opposite. They might incorrectly set up the equation as \(\mathrm{\frac{5}{13} = \frac{BC}{52}}\) instead of \(\mathrm{\frac{5}{13} = \frac{AC}{52}}\).

This leads them to find \(\mathrm{BC = 20}\) first, then \(\mathrm{AC = 48}\), which still gives the correct area of 480. However, in more complex problems, this confusion about adjacent vs. opposite sides can lead to serious errors.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when finding BC using the Pythagorean theorem. They might incorrectly calculate \(\mathrm{52^2 - 20^2}\) or make an error when taking the square root of 2304.

Common mistakes include getting \(\mathrm{BC = 32}\) (from \(\mathrm{\sqrt{1024}}\) instead of \(\mathrm{\sqrt{2304}}\)) or \(\mathrm{BC = 44}\) (from arithmetic errors), leading them to calculate areas like 320 or 440, which don't match any answer choice. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can correctly set up trigonometric relationships and execute multi-step calculations. The key insight is recognizing that cosine gives you the adjacent side, not the opposite side, and then systematically using the Pythagorean theorem to complete the solution.