In the diagram below, $O$ is the circumcenter of $\triangle ABC$. If $\angle BAC=28^\circ$ and $\angle OAC=32^\circ$, then what is the degree measure of $\angle AOC$?

Given that $ O $ is the circumcenter of $ \triangle ABC $, the point $ O $ is the center of the circumcircle of $ \triangle ABC $. Therefore, the segments $ OA $, $ OB $, and $ OC $ are all radii of the circumcircle, making $ \triangle OAC $ an isosceles triangle with $ OA = OC $.

We are tasked with finding $ \angle AOC $, which is the angle at the circumcenter subtended by arc $ AC $ of the circumcircle.

### Step 1: Relationship between the angles
We know that the angle at the circumcenter subtended by an arc is twice the angle subtended at the circumference by the same arc. Thus, the relationship between the angles is:

\[
\angle AOC = 2 \times \angle BAC
\]

From the problem, we are given that $ \angle BAC = 28^\circ $.

### Step 2: Calculate $ \angle AOC $
Using the relationship, we can now compute:

\[
\angle AOC = 2 \times 28^\circ = 56^\circ
\]

Thus, the degree measure of $ \angle AOC $ is $ \boxed{56^\circ} $.

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:
1. What is the circumcenter of a triangle, and how is it located geometrically?
2. How do the properties of isosceles triangles help in solving problems with circumcenters?
3. Can you prove why the angle subtended at the circumcenter is twice the angle subtended at the vertex?
4. How does the concept of circumcircles apply to non-right triangles?
5. What are some applications of the circumcenter in real-world geometry?

**Tip:** Remember that angles subtended by the same arc in a circle have special relationships. The central angle is always double the angle on the circumference.

In the diagram below, $O$ is the circumcenter of $\triangle ABC$. If $\angle BAC=28^\circ$ and $\angle OAC=32^\circ$, then what is the degree measure of $\angle AOC$?

Learn how to solve for angle AOC in a triangle with circumcenter, using the relationship between central and inscribed angles. This problem explores geometric properties of triangles and circles, applying key theorems like the Circumcenter and Isosceles Triangle Theorem. Suitable for Grades 9-12.

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