The side lengths of right triangle RST are given. Triangle RST is similar to triangle UVW, where S corresponds to...

1. TRANSLATE the problem information

• Given information:
  • Right triangle RST with sides \(\mathrm{RS = 440}\), \(\mathrm{ST = 384}\), \(\mathrm{TR = 584}\)
  • Triangle RST is similar to triangle UVW
  • S corresponds to V, T corresponds to W (so R corresponds to U)
• What we need: \(\mathrm{tan W}\)

2. INFER which angle is the right angle

• In any right triangle, the hypotenuse is always the longest side
• Comparing our sides: \(\mathrm{RS = 440}\), \(\mathrm{ST = 384}\), \(\mathrm{TR = 584}\)
• \(\mathrm{TR = 584}\) is the longest, so TR is the hypotenuse
• The right angle is always opposite the hypotenuse, so angle S is the right angle

3. INFER the connection between triangles

• Since the triangles are similar with T corresponding to W:
• Corresponding angles in similar triangles are equal
• Therefore: \(\mathrm{tan W = tan T}\)

4. APPLY the tangent ratio definition

• For angle T in triangle RST:
  • The opposite side (across from angle T) is \(\mathrm{RS = 440}\)
  • The adjacent side (next to angle T, not the hypotenuse) is \(\mathrm{ST = 384}\)
• \(\mathrm{tan T = \frac{opposite}{adjacent} = \frac{RS}{ST} = \frac{440}{384}}\)

5. SIMPLIFY the fraction

• \(\mathrm{\frac{440}{384} = \frac{55 \times 8}{48 \times 8} = \frac{55}{48}}\)
• Therefore: \(\mathrm{tan W = \frac{55}{48}}\)

Answer: D. \(\mathrm{\frac{55}{48}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students struggle to identify which angle is the right angle in the triangle. They might assume angle R or angle T is the right angle without using the key insight that the longest side is always the hypotenuse. This leads to using the wrong sides in their tangent ratio calculation and selecting incorrect answers like Choice A (\(\mathrm{\frac{48}{73}}\)) or Choice B (\(\mathrm{\frac{55}{73}}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand the triangle correspondence, thinking that since S↔V, they need to find tan V instead of tan W. Or they confuse which sides are opposite and adjacent to the angle they're working with. This leads to calculating tan⁻¹ (reciprocal of tangent) and selecting Choice C (\(\mathrm{\frac{48}{55}}\)).

The Bottom Line:

This problem requires students to work backwards from side lengths to identify triangle orientation, then forward through similarity to find the desired trigonometric ratio. The key insight is recognizing that the longest side determines everything else about the triangle's structure.