Answer :
To find the area of sector [tex]\( AOB \)[/tex] given that points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are on a circle centered at [tex]\( O \)[/tex], we can follow these steps:
1. Identify the Radius:
The problem states that [tex]\( OA = 5 \)[/tex]. Therefore, the radius of the circle is 5 units.
2. Understand the Ratio for the Arc:
The length of arc [tex]\( AB \)[/tex] is given as [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle. This means [tex]\( AB \)[/tex] is one-quarter of the total circumference.
3. Calculate the Circumference of the Circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \times \pi \times \text{radius}
\][/tex]
Using [tex]\(\pi = 3.14\)[/tex] and the radius being 5, the circumference is:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
4. Calculate the Length of Arc [tex]\( AB \)[/tex]:
Since the arc [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the length of arc [tex]\( AB \)[/tex] is:
[tex]\[
\text{Arc length} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
5. Calculate the Area of the Whole Circle:
The area of a circle is given by:
[tex]\[
\text{Area} = \pi \times \text{radius}^2
\][/tex]
So the area of the circle is:
[tex]\[
\text{Area} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
6. Calculate the Area of Sector [tex]\( AOB \)[/tex]:
The area of a sector relative to the circle is the same as the proportion of the arc length to the circumference. Since arc [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the area of sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle:
[tex]\[
\text{Sector area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
7. Choose the Closest Answer:
From the options given, [tex]\( 19.625 \)[/tex] is closest to [tex]\( 19.6 \)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{19.6} \text{ square units}
\][/tex]
1. Identify the Radius:
The problem states that [tex]\( OA = 5 \)[/tex]. Therefore, the radius of the circle is 5 units.
2. Understand the Ratio for the Arc:
The length of arc [tex]\( AB \)[/tex] is given as [tex]\(\frac{1}{4}\)[/tex] of the circumference of the circle. This means [tex]\( AB \)[/tex] is one-quarter of the total circumference.
3. Calculate the Circumference of the Circle:
The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \times \pi \times \text{radius}
\][/tex]
Using [tex]\(\pi = 3.14\)[/tex] and the radius being 5, the circumference is:
[tex]\[
\text{Circumference} = 2 \times 3.14 \times 5 = 31.4 \text{ units}
\][/tex]
4. Calculate the Length of Arc [tex]\( AB \)[/tex]:
Since the arc [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the length of arc [tex]\( AB \)[/tex] is:
[tex]\[
\text{Arc length} = \frac{1}{4} \times 31.4 = 7.85 \text{ units}
\][/tex]
5. Calculate the Area of the Whole Circle:
The area of a circle is given by:
[tex]\[
\text{Area} = \pi \times \text{radius}^2
\][/tex]
So the area of the circle is:
[tex]\[
\text{Area} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]
6. Calculate the Area of Sector [tex]\( AOB \)[/tex]:
The area of a sector relative to the circle is the same as the proportion of the arc length to the circumference. Since arc [tex]\( AB \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the circumference, the area of sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle:
[tex]\[
\text{Sector area} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]
7. Choose the Closest Answer:
From the options given, [tex]\( 19.625 \)[/tex] is closest to [tex]\( 19.6 \)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{19.6} \text{ square units}
\][/tex]
