https://file.aiquickdraw.com/mathbot/file/1726121996733w2s3abie.jpg

Let's go through the questions one by one from the image you've uploaded.

### 1. Given:
- The arc length \( BC = 12 \, \text{cm} \).
- The angle \( \angle ABC = 72^\circ \) and \( \angle BCD = 48^\circ \).

To find the arc length \( AD \), we'll use the proportionality of the arc length to the central angle:
\[
\frac{\text{Arc length}}{\text{Total circumference}} = \frac{\text{Central angle}}{360^\circ}
\]
The total circumference is calculated as \( 2\pi r \).

### 2. The area of the sector \( AOB \) is given as \( 105 \, \text{cm}^2 \). To find the area of sector \( COD \), we apply the area of a sector formula:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
The angle of sector \( AOB = 120^\circ \) and sector \( COD = 56^\circ \).

### 3. For this question, it involves solving for the radius \( r \) using the known arc lengths and the relationship between chord lengths and arcs.

### 4. You're asked to find the area of the sector \( COD \) using known angles and the area of sector \( AOB \).

### 5. This question involves calculating the arc length \( PQ \) using the sum of the angles and the provided distances of the arcs and chords.

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Let me know if you'd like further clarification on any specific question or a step-by-step solution! Here are some related questions for practice:

1. How do you calculate the length of an arc given the radius and the angle in degrees?
2. What is the formula for the area of a sector of a circle?
3. How can the radius of a circle be derived from known arc lengths and angles?
4. What is the relationship between the circumference of a circle and its radius?
5. How do chord lengths relate to their corresponding arc lengths in a circle?

**Tip:** Always remember that for sectors, the area and arc length are directly proportional to their central angles.

Multiple geometry problems involving circle sectors, arc lengths, and angles.

Solve problems involving sector area and arc length using proportionality and circle geometry theorems. Learn how to apply formulas to find arc lengths and areas with different angles. Suitable for Grades 9-11 students.

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