In triangle XYZ, angle Y is a right angle, the measure of angle Z is 33°, and the length of...

1. TRANSLATE the problem information

• Given information:
  • Triangle XYZ with right angle at Y
  • Angle \(\mathrm{Z = 33°}\)
  • Side \(\mathrm{YZ = 26\, units}\)
  • Area can be expressed as \(\mathrm{k \tan 33°}\)
• What this tells us: We have a right triangle where we know one angle and one side, need to find a constant k

2. INFER the approach

• Since we have a right triangle with one known angle and adjacent side, we can use trigonometry
• Strategy: Use tangent to find the unknown leg, then calculate area
• The tangent ratio will help us find XY using the known information

3. TRANSLATE the trigonometric setup

• In right triangle XYZ with right angle at Y:
  • YZ (length 26) is adjacent to angle Z
  • XY is opposite to angle Z
  • \(\mathrm{\tan 33° = \frac{opposite}{adjacent} = \frac{XY}{YZ}}\)

4. SIMPLIFY to find the unknown leg

• Substitute known values: \(\mathrm{\tan 33° = \frac{XY}{26}}\)
• Solve for XY: \(\mathrm{XY = 26 \tan 33°}\)

5. APPLY the area formula

• \(\mathrm{Area\, of\, right\, triangle = \frac{1}{2} \times base \times height}\)
• \(\mathrm{Area = \frac{1}{2} \times YZ \times XY}\)
• \(\mathrm{Area = \frac{1}{2} \times 26 \times (26 \tan 33°)}\)

6. SIMPLIFY the area expression

• \(\mathrm{Area = \frac{1}{2} \times 26² \times \tan 33°}\)
• \(\mathrm{Area = \frac{1}{2} \times 676 \times \tan 33°}\)
• \(\mathrm{Area = 338 \tan 33°}\)

7. INFER the final answer

• Since area = \(\mathrm{k \tan 33°}\) and we found area = \(\mathrm{338 \tan 33°}\)
• Therefore: \(\mathrm{k = 338}\)

Answer: 338

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often confuse which side is opposite vs. adjacent to the given angle, leading them to set up \(\mathrm{\tan 33° = \frac{26}{XY}}\) instead of \(\mathrm{\tan 33° = \frac{XY}{26}}\).

This gives them \(\mathrm{XY = \frac{26}{\tan 33°}}\), making the area = \(\mathrm{\frac{1}{2} \times 26 \times \frac{26}{\tan 33°} = \frac{338}{\tan 33°}}\). They might then think \(\mathrm{k = 338}\) but with incorrect reasoning, or get confused about how to match this with \(\mathrm{k \tan 33°}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the approach but make arithmetic errors when calculating \(\mathrm{\frac{1}{2} \times 26² = \frac{1}{2} \times 676}\), getting values like 676 or other incorrect calculations for k.

This leads to confusion about the final answer since their algebraic setup was correct but computational errors threw off the result.

The Bottom Line:

This problem challenges students to correctly identify trigonometric relationships in a right triangle context and then carefully execute multi-step algebraic simplification to isolate the desired constant.