Math Problem Statement

The size of ∠OBC is

The size of ∠OBC is

Solution

To determine the size of OBC\angle OBC, we can use the following approach:

  1. Since OO is the center of the circle, OA\overline{OA}, OB\overline{OB}, and OC\overline{OC} are radii of the circle.
  2. We are given that AOC=120\angle AOC = 120^\circ, which is the angle at the center.

Now, let's apply the angle subtended by the same arc theorem, which states that the angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at the circumference.

  • Arc ACAC subtends AOC\angle AOC at the center, which is 120120^\circ.
  • Therefore, the angle subtended by arc ACAC at point BB on the circumference, ABC\angle ABC, will be half of AOC\angle AOC.

ABC=12×120=60\angle ABC = \frac{1}{2} \times 120^\circ = 60^\circ

Thus, OBC\angle OBC is 60°.

Let me know if you need more details or have further questions!

Here are 5 related questions for deeper understanding:

  1. What is the measure of ABC\angle ABC in this circle?
  2. How can you prove that OBC\triangle OBC is isosceles?
  3. What is the total sum of the interior angles in OBC\triangle OBC?
  4. How would you calculate the length of OC\overline{OC} if the radius of the circle is known?
  5. How does the property of cyclic quadrilaterals apply to this diagram?

Tip: Always remember that angles at the center of a circle are twice the angles on the circumference subtended by the same arc.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Angles at the center and circumference

Formulas

Angle at center = 2 × Angle at circumference

Theorems

Angle subtended by the same arc theorem

Suitable Grade Level

Grades 9-11