https://file.aiquickdraw.com/mathbot/file/1729112618485z6re7tc8.jpg

Let's break down the problem from the image.

### Problem 2: Find the shaded area when:
For this, we are working with sectors of a circle. The formula to find the area of a sector is:

\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]

Where:
- \( \theta \) is the central angle in degrees.
- \( r \) is the radius of the circle.

#### 2(a) \( r = 6 \, \text{cm}, \, \theta = 70^\circ \)

\[
\text{Area} = \frac{70^\circ}{360^\circ} \times \pi \times (6)^2
\]
\[
\text{Area} = \frac{70}{360} \times \pi \times 36
\]
\[
\text{Area} = \frac{7}{36} \times \pi \times 36 = 7 \pi \, \text{cm}^2 \approx 21.99 \, \text{cm}^2
\]

#### 2(b) \( r = 14 \, \text{cm}, \, \theta = 104^\circ \)

\[
\text{Area} = \frac{104^\circ}{360^\circ} \times \pi \times (14)^2
\]
\[
\text{Area} = \frac{104}{360} \times \pi \times 196
\]
\[
\text{Area} = \frac{26}{90} \times \pi \times 196 \approx 177.85 \, \text{cm}^2
\]

#### 2(c) \( r = 5 \, \text{cm}, \, \theta = 80^\circ \)

\[
\text{Area} = \frac{80^\circ}{360^\circ} \times \pi \times (5)^2
\]
\[
\text{Area} = \frac{80}{360} \times \pi \times 25
\]
\[
\text{Area} = \frac{2}{9} \times \pi \times 25 \approx 17.45 \, \text{cm}^2
\]

### Problem 3: Find \( \theta \) and the shaded area when:
For this problem, the formula for calculating the angle \( \theta \) can be derived from the formula for the arc length, which is:

\[
\text{Arc length} = \frac{\theta}{360^\circ} \times 2 \pi r
\]

#### 3(a) \( AB = 10 \, \text{cm}, \, r = 10 \, \text{cm} \)

Here, \( AB \) is the arc length, and \( r \) is the radius.

\[
10 = \frac{\theta}{360^\circ} \times 2 \pi \times 10
\]
\[
10 = \frac{\theta}{360^\circ} \times 20 \pi
\]
\[
\theta = \frac{10 \times 360}{20 \pi} \approx 57.30^\circ
\]

Now, find the shaded area:

\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{57.30}{360} \times \pi \times (10)^2
\]
\[
\text{Area} \approx \frac{57.30}{360} \times \pi \times 100 = 50 \, \text{cm}^2
\]

#### 3(b) \( AB = 8 \, \text{cm}, \, r = 5 \, \text{cm} \)

\[
8 = \frac{\theta}{360^\circ} \times 2 \pi \times 5
\]
\[
8 = \frac{\theta}{360^\circ} \times 10 \pi
\]
\[
\theta = \frac{8 \times 360}{10 \pi} \approx 91.66^\circ
\]

Now, find the shaded area:

\[
\text{Area} = \frac{91.66}{360} \times \pi \times (5)^2
\]
\[
\text{Area} \approx \frac{91.66}{360} \times \pi \times 25 = 20.01 \, \text{cm}^2
\]

Let me know if you'd like further details or clarifications!

Here are some follow-up questions for practice:

1. What happens to the area of a sector if the radius is doubled?
2. How do you calculate the length of an arc if the area of the sector is known?
3. Can the area of a sector ever be greater than the area of the circle itself? Why or why not?
4. What is the formula for the circumference of a circle, and how does it relate to sectors?
5. If the angle \( \theta \) becomes 180°, what is the name of the corresponding geometric shape?

**Tip**: Always keep track of your units, especially when dealing with areas (square units) and lengths (linear units).

Find the shaded area and the angle θ for various values of r and AB in a circle sector.

Explore how to find the shaded area and the angle in a circle sector with given values of radius (r) and arc length (AB). This problem covers key geometric concepts, including the area of sectors and arc lengths. Suitable for students in Grades 8-10.

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