High School

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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 } \widehat{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 :

The circle has a radius of [tex]$5$[/tex], so its area is calculated by

[tex]$$
\text{Circle Area} = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5.
$$[/tex]

Since the arc length [tex]$\widehat{AB}$[/tex] represents [tex]$\frac{1}{4}$[/tex] of the entire circumference, the corresponding sector occupies [tex]$\frac{1}{4}$[/tex] of the whole circle. Therefore, the area of sector [tex]$AOB$[/tex] is

[tex]$$
\text{Sector Area} = \frac{1}{4} \times 78.5 = 19.625.
$$[/tex]

Rounded to one decimal place, the area is approximately [tex]$19.6$[/tex] square units.

Thus, the correct answer is [tex]$\boxed{19.6}$[/tex] square units.

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