Points A, B, and C lie on a circle with center O and a radius of 10. Together with center...

1. TRANSLATE the problem information

• Given information:
  • Points A, B, C are on a circle with center O and radius 10
  • These points form quadrilateral OABC with center O
  • Length AB equals length BC
  • Perimeter of quadrilateral OABC is 34

• What this tells us: We need to find the lengths of all four sides to work with the perimeter

2. INFER which sides we already know

• Since A and C are on the circle and O is the center, segments OA and OC are both radii
• Therefore: \(\mathrm{OA = 10}\) and \(\mathrm{OC = 10}\)
• We're told \(\mathrm{AB = BC}\), so we can call both of these unknown lengths \(\mathrm{x}\)

3. TRANSLATE the perimeter constraint into an equation

• Perimeter = sum of all four sides
• Perimeter = \(\mathrm{OA + AB + BC + CO = 10 + x + x + 10 = 20 + 2x}\)
• Since perimeter equals 34: \(\mathrm{20 + 2x = 34}\)

4. SIMPLIFY to solve for x

• \(\mathrm{20 + 2x = 34}\)
• \(\mathrm{2x = 34 - 20 = 14}\)
• \(\mathrm{x = 14 ÷ 2 = 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that OA and OC are radii because they don't carefully track which points are on the circle versus which is the center.

They might assume all four sides are unknown and set up: \(\mathrm{x + y + z + w = 34}\), leading to confusion since they can't solve for four unknowns with one equation. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{20 + 2x = 34}\) but make an algebra error, such as getting \(\mathrm{2x = 54}\) instead of \(\mathrm{2x = 14}\), leading to \(\mathrm{x = 27}\).

Since 27 isn't among the answer choices, this causes them to get stuck and randomly select an answer.

The Bottom Line:

The key insight is recognizing that some sides of the quadrilateral are radii (and therefore known), while others are chords (unknown). Students must carefully track which points lie on the circle to identify the radii.