https://file.aiquickdraw.com/mathbot/file/1727935698965k0h8gua2.jpg

The problem you've shared from the image involves a triangle \( \triangle ABC \), where:

- The angle at vertex \( A \) is \( 48^\circ \).
- \( I \) is the incenter (the center of the inscribed circle) of the triangle.
- You are asked to determine the angle \( \angle BIC \).

### Key Insight:
In any triangle, the angle subtended by the incenter (point \( I \)) at two vertices \( B \) and \( C \) of the triangle is given by the formula:

\[
\angle BIC = 90^\circ + \frac{\angle A}{2}
\]

where \( \angle A \) is the angle at vertex \( A \).

### Solution:
Since \( \angle A = 48^\circ \), we substitute this into the formula:

\[
\angle BIC = 90^\circ + \frac{48^\circ}{2}
\]
\[
\angle BIC = 90^\circ + 24^\circ = 114^\circ
\]

Thus, \( \angle BIC = 114^\circ \).

Would you like further explanation or additional details on any step?

### Follow-up questions:
1. What is the role of the incenter in a triangle, and how does it relate to angle calculations?
2. How do we derive the formula \( \angle BIC = 90^\circ + \frac{\angle A}{2} \)?
3. Can the formula for \( \angle BIC \) be applied to any type of triangle?
4. How does the concept of the incenter relate to other triangle centers like the centroid or circumcenter?
5. If \( \angle A \) were changed, how would \( \angle BIC \) be affected?

### Tip:
For problems involving triangle centers like the incenter or circumcenter, memorizing key angle relationships can significantly speed up solving geometric problems.

In triangle ABC, the angle at A is 48°. If I is the center of the inscribed circle of ABC, how many degrees is ∠BIC?

This problem involves calculating the angle ∠BIC in triangle ABC, where the angle at vertex A is 48°, and I is the incenter. Learn how to apply the angle at the incenter theorem to find ∠BIC using the formula ∠BIC = 90° + (∠A / 2). Suitable for students in Grades 10-12.

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