https://file.aiquickdraw.com/mathbot/file/1727270207078usz7kjvh.jpg

To solve this problem, we need to analyze the geometric relationship between the square and the hexagon. Here's the step-by-step approach:

### Step 1: Identify Key Elements
- \( A, B, C, D \) are the vertices of the square.
- \( O \) is the center of the square.
- \( OC \) is the side of the regular hexagon, and the question asks for the measure of angle \( \alpha \) (between line segments \( OA \) and \( OD \)).

### Step 2: Properties of the Figures
- The square has all right angles, so the angle between \( OA \) and \( OC \) is \( 90^\circ \) (since \( O \) is the center and \( OA \) and \( OC \) are radii aligned with the square's symmetry).
- A regular hexagon inscribed in a circle means that each internal angle between two adjacent sides from the center is \( 60^\circ \). Since \( OC \) is a side of the hexagon, the angle between any adjacent radius (such as \( OC \) and any other radius of the hexagon) will be \( 60^\circ \).

### Step 3: Analyze Angle \( \alpha \)
- The angle \( \alpha \) is the difference between the \( 90^\circ \) (the angle formed by \( OA \) and \( OC \)) and the angle contributed by the hexagon's structure.
- The hexagon causes the angle to widen, adding another portion of an angle formed by the hexagon. Specifically, since \( OD \) is one of the radii of the hexagon and adjacent to \( OC \), the angle between \( OC \) and \( OD \) is \( 60^\circ \).

Thus, angle \( \alpha \) is the sum of \( 90^\circ \) (from the square) and \( 60^\circ \) (from the hexagon):

\[
\alpha = 90^\circ + 30^\circ = 120^\circ.
\]

### Conclusion:
The measure of angle \( \alpha \) is \( 120^\circ \), so the correct answer is **(D) 120°**.

Would you like more details on any of the steps or the reasoning?

### Related Questions:
1. What is the sum of the interior angles of a regular hexagon?
2. How do you find the internal angles of any polygon given the number of sides?
3. Can you calculate the diagonal length of the square if the side is known?
4. What is the relationship between the side lengths of a regular hexagon and its radius?
5. How can we derive the interior angle of a regular polygon?

**Tip:** Remember, for any regular polygon, the central angle formed by radii to two adjacent vertices is \( \frac{360^\circ}{n} \), where \( n \) is the number of sides.

Martin draws a square with vertices A, B, C, D and a regular hexagon with side OC, where O is the center of the square. What is the measure of angle α?

This problem explores the relationship between a square and a regular hexagon. The key challenge is finding the measure of angle α formed by the square and hexagon, using properties of polygons and angle relationships. Suitable for high school geometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation