In isosceles trapezoid PQRS, the parallel sides PQ and RS have lengths 36 and 108, respectively. Point T lies on...

1. TRANSLATE the position information

• Given information:
  • \(\mathrm{PT = 2TS}\) (T divides PS in ratio 2:1)
  • \(\mathrm{PT + TS = PS}\) (T lies on segment PS)

• What this tells us about T's position:
  • \(\mathrm{2TS + TS = PS}\)
  • \(\mathrm{3TS = PS}\)
  • T is located \(\mathrm{\frac{2}{3}}\) of the distance from P toward S

2. INFER the key geometric relationship

• In trapezoids, any segment parallel to the bases has a length that varies linearly between the base lengths
• Since TU is parallel to both PQ (length 36) and RS (length 108), its length depends on how far along T is positioned
• T being \(\mathrm{\frac{2}{3}}\) of the way from P to S means TU is \(\mathrm{\frac{2}{3}}\) of the way from length 36 to length 108

3. SIMPLIFY using linear interpolation

• Formula: \(\mathrm{TU = PQ + (position\:fraction)(RS - PQ)}\)
• \(\mathrm{TU = 36 + \frac{2}{3}(108 - 36)}\)
• \(\mathrm{TU = 36 + \frac{2}{3}(72)}\)
• \(\mathrm{TU = 36 + 48}\)
• \(\mathrm{TU = 84}\)

Answer: C) 84

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the linear interpolation relationship for parallel segments in trapezoids. Instead, they might try to:

• Average the two base lengths: \(\mathrm{\frac{36 + 108}{2} = 72}\)
• This leads them to select Choice B (72)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret \(\mathrm{PT = 2TS}\) as meaning T is at the midpoint, rather than working through the algebra to find T is \(\mathrm{\frac{2}{3}}\) of the way along PS. This causes confusion about where exactly to place the parallel segment, leading to guessing among the answer choices.

The Bottom Line:

The key insight is recognizing that parallel segments in trapezoids don't just average the base lengths - their position matters, and the relationship is linear based on how far along the non-parallel sides you are.