A circle has center O, and points R and S lie on the circle. In triangle ORS, the measure of...

1. TRANSLATE the problem information

• Given information:
  • O is the center of a circle
  • Points R and S lie on the circle
  • In triangle ORS, \(\angle\mathrm{ROS} = 88°\)
  • Need to find \(\angle\mathrm{RSO}\)

2. INFER the key geometric relationship

• Since R and S are on the circle and O is the center, OR and OS are both radii
• All radii of a circle are equal, so \(\mathrm{OR} = \mathrm{OS}\)
• When two sides of a triangle are equal, it's an isosceles triangle
• In isosceles triangles, the base angles (angles opposite the equal sides) are equal

3. INFER which angles are equal

• The equal sides are OR and OS
• The angles opposite these equal sides are \(\angle\mathrm{RSO}\) and \(\angle\mathrm{ORS}\)
• Therefore: \(\angle\mathrm{RSO} = \angle\mathrm{ORS}\)

4. SIMPLIFY using the triangle angle sum

• Let \(\mathrm{x}\) = measure of \(\angle\mathrm{RSO}\)
• Since \(\angle\mathrm{RSO} = \angle\mathrm{ORS}\), both angles measure x°
• Triangle angle sum: \(\angle\mathrm{ROS} + \angle\mathrm{ORS} + \angle\mathrm{RSO} = 180°\)
• Substitute: \(88° + \mathrm{x} + \mathrm{x} = 180°\)
• Combine: \(88° + 2\mathrm{x} = 180°\)
• Solve: \(2\mathrm{x} = 92°\), so \(\mathrm{x} = 46°\)

Answer: 46

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that \(\mathrm{OR} = \mathrm{OS}\) creates an isosceles triangle with special angle properties.

Many students see the triangle ORS and think they need to use complex trigonometry or don't realize the significance of O being the center. They might try to apply general triangle solving methods without recognizing the isosceles property. This leads to confusion and guessing since they lack a systematic approach.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students recognize the isosceles triangle but make arithmetic errors when solving \(2\mathrm{x} + 88 = 180\).

They might incorrectly subtract (getting \(2\mathrm{x} = 88\) instead of \(2\mathrm{x} = 92\)) or divide incorrectly, leading to wrong angle measures like 44° or other values that don't match the correct answer of 46°.

The Bottom Line:

The key insight is connecting the circle's geometric properties (equal radii) to triangle properties (isosceles with equal base angles). Without this connection, students can't systematically approach the problem and resort to guessing.