The area of a sector of a circle is 77 sq cm and the angle of the sector is `45^(@)`. Find the radius of the circle.

Options

A. 7 cm
B. 14 cm
C. 21 cm
D. 28 cm

Answer

To find the radius of the circle given the area of a sector and the angle of the sector, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Area of the sector (A) = 77 sq cm
- Angle of the sector (θ) = 45 degrees

2. **Use the Formula for the Area of a Sector:**
The formula for the area of a sector is given by:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( r \) is the radius of the circle.

3. **Substitute the Given Values into the Formula:**
Substitute \( A = 77 \) and \( \theta = 45 \) into the formula:
\[ 77 = \frac{45}{360} \times \pi r^2 \]

4. **Simplify the Equation:**
First, simplify \( \frac{45}{360} \):
\[ \frac{45}{360} = \frac{1}{8} \]
So the equation becomes:
\[ 77 = \frac{1}{8} \times \pi r^2 \]

5. **Multiply Both Sides by 8:**
To eliminate the fraction, multiply both sides by 8:
\[ 77 \times 8 = \pi r^2 \]
\[ 616 = \pi r^2 \]

6. **Substitute the Value of π:**
Use \( \pi \approx \frac{22}{7} \):
\[ 616 = \frac{22}{7} r^2 \]

7. **Multiply Both Sides by 7:**
To eliminate the fraction, multiply both sides by 7:
\[ 616 \times 7 = 22 r^2 \]
\[ 4312 = 22 r^2 \]

8. **Divide by 22:**
Now, divide both sides by 22 to solve for \( r^2 \):
\[ r^2 = \frac{4312}{22} \]
\[ r^2 = 196 \]

9. **Take the Square Root:**
Finally, take the square root of both sides to find \( r \):
\[ r = \sqrt{196} \]
\[ r = 14 \text{ cm} \]

### Final Answer:
The radius of the circle is \( r = 14 \) cm.

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