Question

in the figure, angle zyx is measured in degrees. the area of the shaded sector can be determined using the formula \\(\frac{m\angle zyx}{360^\circ}(\pi r^2)\\). image of a circle with center y, points x and z, radius r, shaded sector xyz which best explains the formula? \\(\bigcirc\\) the central angle measure of the sector divided by the total angle measure of a circle multiplied by the area of the circle will yield the area of the sector. \\(\bigcirc\\) the central angle measure of the sector divided by the total angle measure of a circle multiplied by the circumference of the circle will yield the area of the sector. \\(\bigcirc\\) the central angle measure of the sector multiplied by the area of the circle will yield the area of the sector. \\(\bigcirc\\) the central angle measure of the sector multiplied by the circumference of the circle will yield the area of the sector.

Explanation:

1. Recall the formula for the area of a sector: A sector is a portion of a circle. The total angle around a circle is \( 360^\circ \). The fraction of the circle that the sector represents is given by the ratio of the central angle of the sector (\( m\angle ZYX \)) to \( 360^\circ \) (the total angle of the circle).
2. The area of the entire circle is \( \pi r^2 \). To find the area of the sector, we multiply this fraction (\( \frac{m\angle ZYX}{360^\circ} \)) by the area of the circle (\( \pi r^2 \)).
3. Analyze the options:

• Option 1: Matches the above reasoning. The central angle of the sector divided by \( 360^\circ \) (total angle of circle) times the area of the circle (\( \pi r^2 \)) gives the sector area.
• Option 2: Circumference is used for arc length, not area of sector. So this is incorrect.
• Option 3: We divide the central angle by \( 360^\circ \), not multiply directly by the circle's area. So this is incorrect.
• Option 4: Circumference is for arc length, and multiplying by central angle doesn't give sector area. Incorrect.

Answer:

The central angle measure of the sector divided by the total angle measure of a circle multiplied by the area of the circle will yield the area of the sector.