In circle TT, mangle, U, T, V, equals, 70, degrees∠UTV=70
∘
and the area of shaded sector = start fraction, 7, divided by, 9, end fraction, pi
9
7
π. Find the length of arc, U, V
UV
⌢
. Express your answer as a fraction times piπ.
To find the length of arc UV, we first need to find the radius of the circle.
Since angle UTV is 70 degrees, we know that it is 70/360 = 7/36 of the full circle. Therefore, the area of the shaded sector is 7/36 of the full circle's area.
Let the radius of the circle be r. Then the area of the shaded sector is 7/36 * πr^2 = 7/9 * πr^2.
Given that the area of the shaded sector is 7/9 * π, we can set up the equation:
7/9 * πr^2 = 7/9 * π
Solving for r, we get:
r^2 = 1
r = 1
Now that we have the radius of the circle, we can find the circumference of the full circle using the formula:
Circumference = 2πr
Circumference = 2π
Since the arc UV is 1/9 of the circumference, we can calculate its length:
Arc UV = (1/9) * 2π = 2/9 * π
Therefore, the length of arc UV is 2/9π.
The answer is 7/9 pi
I apologize for the mistake in my previous response. Let's correct it:
Given that the area of the shaded sector is 7/9 * π, we already found the radius of the circle to be 1.
Now, we need to find the length of arc UV. The formula for the length of an arc in a circle is given by:
Length of arc = (θ/360) * 2πr
Where θ is the central angle in degrees and r is the radius of the circle.
In this case, the central angle θ is 70 degrees and the radius r is 1. Substituting these values into the formula, we get:
Length of arc UV = (70/360) * 2π * 1 = (7/36) * 2π = 7/18 * π
Therefore, the length of arc UV is 7/18 * π, not 7/9 * π as previously calculated. Thank you for pointing out the error.