A circle has a center at point O. Points P and Q are on the circle, forming triangle OPQ. If...

1. TRANSLATE the problem information

• Given information:
  • Circle with center O
  • Points P and Q on the circle
  • Triangle OPQ formed
  • Angle POQ = 104°
  • Need to find angle OQP

2. INFER the key geometric relationship

• Since P and Q are on the circle and O is the center, OP and OQ are both radii
• All radii of a circle are equal, so \(\mathrm{OP} = \mathrm{OQ}\)
• This makes triangle OPQ isosceles with equal sides OP and OQ

3. INFER the angle relationship

• In an isosceles triangle, the angles opposite the equal sides are equal
• Since \(\mathrm{OP} = \mathrm{OQ}\), the angles opposite these sides are equal
• Therefore: \(\angle\mathrm{OPQ} = \angle\mathrm{OQP}\)

4. TRANSLATE into an algebraic equation

• Sum of angles in triangle = 180°
• \(\angle\mathrm{POQ} + \angle\mathrm{OPQ} + \angle\mathrm{OQP} = 180°\)
• Since \(\angle\mathrm{OPQ} = \angle\mathrm{OQP}\), let's call this value x
• So: \(104° + \mathrm{x} + \mathrm{x} = 180°\)

5. SIMPLIFY to find the answer

• \(104° + 2\mathrm{x} = 180°\)
• \(2\mathrm{x} = 180° - 104° = 76°\)
• \(\mathrm{x} = 38°\)

Answer: B) 38

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not recognizing that OP and OQ are radii of the circle, so they must be equal.

Students may treat this as a general triangle and try to use other relationships or get confused about which angles to work with. Without recognizing the isosceles nature of triangle OPQ, they can't apply the crucial property that base angles are equal.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak INFER skill: Confusing which angles are equal in an isosceles triangle.

Some students know it's isosceles but incorrectly think that angle POQ equals one of the base angles, rather than understanding that the two base angles (OPQ and OQP) are equal to each other. They might set up: \(\angle\mathrm{POQ} = \angle\mathrm{OQP}\), leading to \(104° = \angle\mathrm{OQP}\).

This may lead them to select Choice E (104).

The Bottom Line:

This problem requires recognizing the special properties that emerge when a triangle is formed by connecting the center of a circle to two points on the circumference. The key insight is that radii create isosceles triangles.