https://file.aiquickdraw.com/mathbot/file/1730375246004utdm9fzr.jpg

Let's go through each question in the image.

1. **Question 3.2.1:** 
   - **Phrase required:** A line segment that passes through the center of a circle and connects two points on the circumference is known as a **diameter**.

2. **Question 3.2.2:**
   - **Example:** In the diagram, an example of a diameter would be **DG**, since it passes through the center \(O\) and connects two points on the circumference.

3. **Question 3.2.3:** 
   - **Identification of quadrilateral DEFG:** 
     - Quadrilateral **DEFG** is an **inscribed quadrilateral** (or cyclic quadrilateral), as all its vertices lie on the circumference of the circle.
   - **Property of this type of quadrilateral:** In a cyclic quadrilateral, the opposite angles sum up to \(180^\circ\).

4. **Question 3.2.4:** 
   - **Determine the angle DGE:**
     - Since DE is a chord and \( \angle DOE = 45^\circ \) (given at the center), the angle \( \angle DGE \) at the circumference subtended by the same chord DE will be half of \( \angle DOE \).
     - Therefore, \( \angle DGE = \frac{45^\circ}{2} = 22.5^\circ \).

5. **Question 3.2.5:**
   - **Calculate angle \( D_1 \):**
     - \( \angle D_1 \) represents \( \angle ODG \), which is in the right triangle \( \triangle ODG \).
     - Given \( OD = 8 \, \text{cm} \) and \( DG = 12 \, \text{cm} \), we can find \( \angle D_1 \) using trigonometric functions.
     - Using cosine, \( \cos(\angle D_1) = \frac{OD}{DG} = \frac{8}{12} = \frac{2}{3} \).
     - Therefore, \( \angle D_1 = \cos^{-1} \left( \frac{2}{3} \right) \approx 48.19^\circ \).

6. **Question 3.2.6:**
   - **Determine angle \( D_2 \):**
     - Since \( \angle D_1 + \angle D_2 = 90^\circ \) (because they form a right angle at \( O \)), we can calculate \( \angle D_2 \) as follows:
     - \( \angle D_2 = 90^\circ - 48.19^\circ \approx 41.81^\circ \).

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Would you like a detailed breakdown of each calculation? 

Here are five related questions to deepen understanding:

1. What is the significance of a cyclic quadrilateral in a circle?
2. How can you calculate angles subtended by chords at the circumference?
3. Why is the opposite angle property important in cyclic quadrilaterals?
4. How do you use trigonometry to find angles in right triangles?
5. What is the relationship between the central angle and the angle at the circumference?

**Tip:** In cyclic quadrilaterals, remember that opposite angles always sum up to 180 degrees.

Gebruik die diagram van die sirkel, met middelpunt O en vierhoek DEFG, en beantwoord die vrae.

Explore circle geometry with this question involving a cyclic quadrilateral and angles subtended by chords. Learn how to calculate angles using the cyclic quadrilateral theorem, trigonometric functions, and central-inscribed angle relationships. Suitable for high school students (Grades 10-12).

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