Points X and Y lie on a circle with center O. The radius of this circle is 6text{ cm}. Triangle...

1. TRANSLATE the given information

• Given information:
  • Circle has center O and radius 6 cm
  • Points X and Y lie on the circle
  • Triangle OXY has perimeter of 20 cm
  • Need to find length of XY

2. INFER key relationships from circle properties

• Since X and Y are points on the circle with center O:
  • \(\mathrm{OX = radius = 6\text{ cm}}\) (distance from center to any point on circle)
  • \(\mathrm{OY = radius = 6\text{ cm}}\) (distance from center to any point on circle)

This is the crucial insight - triangle OXY has two sides that are automatically known!

3. TRANSLATE the perimeter condition into an equation

• Perimeter = sum of all three sides:
  • \(\mathrm{OX + OY + XY = 20\text{ cm}}\)

4. SIMPLIFY by substituting known values

• Substitute the radius values:
  \(\mathrm{6 + 6 + XY = 20}\)
  \(\mathrm{12 + XY = 20}\)
  \(\mathrm{XY = 20 - 12 = 8}\)

Answer: B (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that OX and OY are both radii of the circle.

Instead, they might think they need to use more complex geometric relationships or formulas. They may try to use the Pythagorean theorem, law of cosines, or other trigonometric relationships without realizing the problem is much simpler. This leads to confusion and guessing since they can't figure out how to proceed without additional angle information.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that \(\mathrm{OX = OY = 6}\), but make an arithmetic error when solving \(\mathrm{12 + XY = 20}\).

Common mistake: \(\mathrm{XY = 12 + 20 = 32}\) (adding instead of subtracting). Since 32 isn't an answer choice, this leads to confusion and random selection.

The Bottom Line:

This problem tests whether students can recognize the fundamental property that all radii of a circle are equal, then apply basic algebra. The geometric setup looks complex, but the solution is surprisingly straightforward once you identify what information the circle property gives you for free.