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 :

We are given that [tex]\(O A = 5\)[/tex] and that the arc [tex]\(\widehat{A B}\)[/tex] represents [tex]\(\frac{1}{4}\)[/tex] of the whole circumference of the circle. The steps to find the area of sector [tex]\(AOB\)[/tex] are as follows:

1. Calculate the area of the entire circle using the formula:
[tex]$$
\text{Area} = \pi r^2
$$[/tex]
Substituting [tex]\(r = 5\)[/tex] and [tex]\(\pi = 3.14\)[/tex]:
[tex]$$
\text{Area} = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \text{ square units}
$$[/tex]

2. Since the arc [tex]\(\widehat{A B}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] of the entire circle's circumference, the central angle corresponding to this arc also represents [tex]\(\frac{1}{4}\)[/tex] of the full circle. Therefore, the area of sector [tex]\(AOB\)[/tex] is:
[tex]$$
\text{Area of sector } AOB = \frac{1}{4} \times \text{Area of circle} = \frac{1}{4} \times 78.5 = 19.625 \text{ square units}
$$[/tex]

3. Rounding [tex]\(19.625\)[/tex] to the closest option, we have approximately [tex]\(19.6\)[/tex] square units.

Thus, the area of sector [tex]\(AOB\)[/tex] is approximately [tex]\(19.6\)[/tex] square units, which corresponds to answer choice:

A. 19.6 square units.

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