Triangle ABC is similar to triangle DEF, where A corresponds to D and C corresponds to F. Angles C and...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC ~ Triangle DEF with A↔D, C↔F
  • Angles C and F are right angles
  • \(\mathrm{tan(A) = \sqrt{3}}\)
  • \(\mathrm{DF = 125}\)
  • Need to find: DE

2. INFER the angle relationship

• Since triangles are similar, corresponding angles are equal
• A corresponds to D, so angle A = angle D
• Therefore: \(\mathrm{tan(D) = tan(A) = \sqrt{3}}\)
• Since \(\mathrm{tan(60°) = \sqrt{3}}\), we know \(\mathrm{angle\hspace{0.2cm}D = 60°}\)

3. INFER the triangle type and find the missing side

• In right triangle DEF: \(\mathrm{angle\hspace{0.2cm}D = 60°}\), \(\mathrm{angle\hspace{0.2cm}F = 90°}\)
• Therefore: \(\mathrm{angle\hspace{0.2cm}E = 180° - 60° - 90° = 30°}\)
• This is a 30-60-90 triangle!

4. SIMPLIFY using the tangent ratio

• In triangle DEF: \(\mathrm{tan(D) = \frac{EF}{DF}}\)
• \(\mathrm{\sqrt{3} = \frac{EF}{125}}\)
• \(\mathrm{EF = 125\sqrt{3}}\)

5. INFER the final answer using special triangle properties

• In a 30-60-90 triangle, sides are in ratio \(\mathrm{1:\sqrt{3}:2}\)
• \(\mathrm{DF = 125}\) (opposite the 30° angle)
• \(\mathrm{EF = 125\sqrt{3}}\) (opposite the 60° angle) ✓
• \(\mathrm{DE = 2 \times 125 = 250}\) (hypotenuse, opposite the 90° angle)

Answer: D. 250

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find \(\mathrm{EF = 125\sqrt{3}}\) correctly but don't recognize this as a 30-60-90 triangle, so they don't know how to find the hypotenuse DE. They might think EF is the final answer since it involves \(\mathrm{\sqrt{3}}\).

This may lead them to select Choice C (\(\mathrm{125\sqrt{3}}\)).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that \(\mathrm{tan(60°) = \sqrt{3}}\), so they can't determine that \(\mathrm{angle\hspace{0.2cm}D = 60°}\). Without knowing the specific angle, they can't use the 30-60-90 triangle properties and get stuck trying to use only the tangent ratio.

This leads to confusion and guessing.

The Bottom Line:

This problem requires recognizing the connection between \(\mathrm{tan(\sqrt{3})}\) and 60° angles, then applying special right triangle properties. Students who only focus on the tangent calculation miss the elegant geometric insight that makes the solution straightforward.