An isosceles trapezoid is inscribed in a semicircle such that the longer base of the trapezoid lies along the diameter...

1. TRANSLATE the problem information

• Given information:
  • Isosceles trapezoid inscribed in semicircle
  • Longer base lies along diameter
  • Non-parallel sides = radius length
  • Area = \(\mathrm{147√3}\) square units
  • Find: diameter length

• What this tells us: If radius = \(\mathrm{r}\), then diameter = \(\mathrm{2r}\) and non-parallel sides = \(\mathrm{r}\)

2. VISUALIZE the geometric setup

• VISUALIZE the semicircle with center O and the trapezoid inside it
• Place the diameter horizontally with endpoints \(\mathrm{A(-r, 0)}\) and \(\mathrm{B(r, 0)}\)
• The trapezoid's upper vertices C and D lie on the semicircle
• The key insight: \(\mathrm{AC = BD = r}\) (given), and \(\mathrm{OC = OD = r}\) (radii)

3. INFER the triangle relationships

• Since triangle AOC has sides \(\mathrm{OA = OC = AC = r}\), it's equilateral
• Similarly, triangle BOD is equilateral
• Each equilateral triangle has a \(\mathrm{60°}\) central angle
• The remaining angle \(\mathrm{COD = 180° - 60° - 60° = 60°}\)
• Therefore, triangle COD is also equilateral with all sides = \(\mathrm{r}\)

4. INFER the area calculation strategy

• The trapezoid consists of exactly three equilateral triangles
• Each triangle has side length \(\mathrm{r}\)
• Total area = \(\mathrm{3 × (area\ of\ one\ equilateral\ triangle)}\)

5. SIMPLIFY to find the radius

• Area of equilateral triangle with side \(\mathrm{r}\): \(\frac{\mathrm{r²√3}}{4}\)
• Total trapezoid area: \(\mathrm{3 × \frac{r²√3}{4} = \frac{3r²√3}{4}}\)
• Set equal to given area: \(\frac{\mathrm{3r²√3}}{4} = \mathrm{147√3}\)
• Divide both sides by \(\mathrm{√3}\): \(\frac{\mathrm{3r²}}{4} = \mathrm{147}\)
• Multiply by \(\frac{4}{3}\): \(\mathrm{r² = 196}\)
• Take square root: \(\mathrm{r = 14}\)

6. Find the diameter

• Diameter = \(\mathrm{2r = 2(14) = 28}\)

Answer: C (28)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Students struggle to see how the trapezoid relates to the semicircle geometrically. They may try to use the standard trapezoid area formula \(\mathrm{A = \frac{1}{2}(b₁ + b₂)h}\) but can't figure out the height or the length of the shorter base without visualizing the geometric relationships.

Without the visualization, they get stuck early and may resort to guessing or trying to work backwards from answer choices. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Even students who visualize the setup may not recognize that the configuration creates equilateral triangles. They might attempt complex coordinate geometry or trigonometry instead of seeing the elegant solution through equilateral triangle properties.

This overcomplication leads to getting bogged down in unnecessary calculations and potentially running out of time, causing them to guess from the remaining answer choices.

The Bottom Line:

This problem rewards geometric insight over computational complexity. The key breakthrough is recognizing that the specific constraints create a beautiful pattern of three equilateral triangles, which makes the area calculation straightforward.