High School

Select the correct answer.

Points [tex] A [/tex] and [tex] B [/tex] lie on a circle centered at point [tex] O [/tex]. If [tex] OA = 5 [/tex] and [tex]\frac{\text{length of } \hat{AB}}{\text{circumference}} = \frac{1}{4}[/tex], what is the area of sector [tex] AOB [/tex]? Use the value [tex]\pi = 3.14[/tex], and choose the closest answer.

A. 19.6 square units
B. 39.3 square units
C. 7.85 square units
D. 15.7 square units

Answer :

To find the area of sector [tex]\( AOB \)[/tex], we can follow these steps:

1. Identify the Given Information:
- The radius of the circle, [tex]\( OA \)[/tex], is [tex]\( 5 \)[/tex] units.
- The fraction of the circumference corresponding to arc [tex]\( \hat{AB} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].

2. Calculate the Total Area of the Circle:
- The formula for the area of a circle is [tex]\(\pi \times \text{radius}^2\)[/tex].
- Using the given radius and [tex]\(\pi = 3.14\)[/tex],
[tex]\[
\text{Area of the circle} = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ square units}
\][/tex]

3. Determine the Area of Sector [tex]\( AOB \)[/tex]:
- Since the arc [tex]\(\hat{AB}\)[/tex] makes up [tex]\(\frac{1}{4}\)[/tex] of the circle's circumference, the area of the sector [tex]\( AOB \)[/tex] is also [tex]\(\frac{1}{4}\)[/tex] of the total area of the circle.
- Calculate the area of the sector:
[tex]\[
\text{Area of sector } AOB = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
\][/tex]

4. Select the Closest Answer:
- Comparing [tex]\( 19.625 \)[/tex] square units with the given options, the closest answer is:
- A. 19.6 square units

Therefore, the area of sector [tex]\( AOB \)[/tex] is approximately [tex]\( 19.6 \)[/tex] square units.

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