In circle TT, mangle, U, T, V, equals, 70, degrees∠UTV=70

Question

∘
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.

Answer 1

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.

Answer 2

The answer is 7/9 pi

Answer 3

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.