https://file.aiquickdraw.com/mathbot/file/17266090994433gcohnao.jpg

The diagram depicts a circle with center \( O \), and several points and lines marked, including \( A \), \( B \), \( C \), and \( D \). Additionally, it refers to angles \( \theta \) and radii \( r \). The image has some related questions, which I'll solve step by step.

### Let's begin with each question:

#### 1. **The length of the straight line AD**:
   - The line \( AD \) is a radius of the circle since \( O \) is the center of the circle. Thus, the length of \( AD \) is the radius of the circle, \( r \).

#### 2. **The area of the sector of the circle (center A) bounded by the lines AB and AD and the arc BCD**:
   - The sector mentioned is centered at \( A \) and subtended by the angle \( \theta \). The formula for the area of a sector is:
     \[
     \text{Area of sector} = \frac{1}{2} r^2 \theta
     \]
     where \( \theta \) is the angle in radians.

#### 3. **The area of the triangle OAD**:
   - The triangle \( OAD \) is a right triangle with angle \( \theta \) at \( A \), and both \( OA \) and \( OD \) are radii of the circle.
     - Area of triangle \( OAD \) is:
     \[
     \text{Area of triangle} = \frac{1}{2} r^2 \sin(\theta)
     \]

#### 4. **The area of the segment ADE**:
   - The segment \( ADE \) is part of the circle minus the triangle \( OAD \). The formula for the area of a circular segment is:
     \[
     \text{Area of segment} = \frac{1}{2} r^2 (\theta - \sin(\theta))
     \]

Would you like detailed calculations or further explanation on any of these steps?

### Relative Questions:
1. How do you calculate the length of an arc on a circle?
2. What is the formula for the circumference of a circle?
3. Can you explain the relationship between the radius and the central angle of a sector?
4. How is the area of a triangle related to its base and height?
5. What is the difference between the area of a sector and the area of a segment?

**Tip:** Always express angles in radians when calculating areas or lengths in circular geometry, as many formulas are derived using radians.

Find the length of the straight line AD, the area of the sector of the circle (centre A) bounded by the lines AB and AD and the arc BCD, the area of the triangle OAD, and the area of the segment ADE.

Learn how to calculate the length of a line, the area of a sector, a triangle, and a segment in a circle. This problem involves key circle geometry concepts and trigonometric formulas, suitable for students in Grades 10-12.

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