In triangle XYZ, angle Z is a right angle and the length of YZ is 24 units. If tan X...

1. TRANSLATE the problem information

• Given information:
  • Triangle XYZ has a right angle at Z
  • \(\mathrm{YZ} = 24\) units
  • \(\tan \mathrm{X} = \frac{12}{35}\)
  • Need to find perimeter

2. INFER the triangle setup and strategy

• Since angle Z is the right angle, sides YZ and XZ are the legs, and XY is the hypotenuse
• For angle X: YZ is the opposite side, XZ is the adjacent side
• Strategy: Use the tangent ratio to find XZ, then Pythagorean theorem for XY

3. TRANSLATE the tangent relationship

• \(\tan \mathrm{X} = \frac{\text{opposite}}{\text{adjacent}} = \frac{\mathrm{YZ}}{\mathrm{XZ}}\)
• Substituting known values: \(\frac{12}{35} = \frac{24}{\mathrm{XZ}}\)

4. SIMPLIFY to find the missing leg

• Cross-multiply: \(12 \times \mathrm{XZ} = 24 \times 35\)
• \(12\mathrm{XZ} = 840\)
• \(\mathrm{XZ} = 840 \div 12 = 70\) units

5. INFER the need for Pythagorean theorem

• Now we need the hypotenuse XY to complete the perimeter
• Use: \(\mathrm{XY}^2 = \mathrm{YZ}^2 + \mathrm{XZ}^2\)

6. SIMPLIFY using Pythagorean theorem

• \(\mathrm{XY}^2 = 24^2 + 70^2\)
• \(\mathrm{XY}^2 = 576 + 4900 = 5476\)
• \(\mathrm{XY} = \sqrt{5476} = 74\) units (use calculator)

7. TRANSLATE perimeter definition

• Perimeter = sum of all three sides
• \(\text{Perimeter} = \mathrm{YZ} + \mathrm{XZ} + \mathrm{XY} = 24 + 70 + 74 = 168\) units

Answer: B. 168

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which sides are opposite and adjacent to angle X, setting up \(\tan \mathrm{X} = \frac{\mathrm{XZ}}{\mathrm{YZ}}\) instead of \(\tan \mathrm{X} = \frac{\mathrm{YZ}}{\mathrm{XZ}}\).

This leads them to solve \(\frac{12}{35} = \frac{\mathrm{XZ}}{24}\), giving \(\mathrm{XZ} = \frac{12 \times 24}{35} \approx 8.2\). When they use Pythagorean theorem with legs of 24 and 8.2, they get a hypotenuse around 25.4, leading to a perimeter around 57.6. This doesn't match any answer choice exactly, causing confusion and guessing.

Second Most Common Error:

Incomplete INFER reasoning: Students correctly find \(\mathrm{XZ} = 70\) but forget they need to find the third side. They add only the two legs: \(24 + 70 = 94\), which doesn't match any answer choice, leading them to guess.

The Bottom Line:

This problem requires careful attention to the trigonometric setup - students must correctly identify which side is opposite and which is adjacent to the given angle, then remember that a complete perimeter requires all three sides of the triangle.