Question

In circle gg, g, h, equals, 2gh=2 and the area of shaded sector = start fraction, 8, divided by, 9, end fraction, pi 9 8 ​ π. find the length of arc, h, i hi ⌢ . express your answer as a fraction times piπ.

Options:
a) (16/9)π
b) (9/8)π
c) (8/9)π
d) (2/3)π

Answer 1

To find the length of arc hi ⌢, we need to first find the measure of the central angle that subtends this arc. From the given information, we know that the shaded sector has an area of (8/9)π. We can use the formula for the area of a sector to determine the measure of the central angle, which is equal to the length of the arc.

Step-by-step explanation:

To find the length of arc hi ⌢, we need to first find the measure of the central angle that subtends this arc. From the given information, we know that the shaded sector has an area of 8⁄9π. We also know that the area of a sector is given by the formula A = θ/360° * πr², where A is the area, θ is the central angle in degrees, and r is the radius. Solving for θ, we have θ = A * 360° / (πr²). Substituting the given values, we get θ = 8⁄9π * 360° / (πr²). Simplifying this expression gives us θ = 320° / r².

Since the central angle θ is also the angle at the center of the circle, it is equal to the measure of arc hi ⌢. So, the length of arc hi ⌢ is 320° times the radius r. The answer should be expressed as a fraction times π, so the final answer is (320° / r)π.