Question

in circle t, tu = 3 and the area of shaded sector = \\(\frac{5}{4}\pi\\). find m\\(\angle utv\\).

answer - attempt 1 out of 3

m\\(\angle utv = \square^\circ\\)

Explanation:

Step1: Recall the sector area formula

The formula for the area of a sector of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360}\times\pi r^{2} \). Here, \( TU \) is the radius, so \( r = 3 \), and the area of the sector \( A=\frac{5}{4}\pi \).

Step2: Substitute the known values into the formula

Substitute \( A=\frac{5}{4}\pi \) and \( r = 3 \) into the sector area formula:

$$ \frac{5}{4}\pi=\frac{\theta}{360}\times\pi\times(3)^{2} $$

Simplify the right - hand side: \( \frac{\theta}{360}\times9\pi=\frac{9\theta\pi}{360}=\frac{\theta\pi}{40} \)

Step3: Solve for \( \theta \)

We have the equation \( \frac{5}{4}\pi=\frac{\theta\pi}{40} \). Since \( \pi
eq0 \), we can divide both sides of the equation by \( \pi \):

$$ \frac{5}{4}=\frac{\theta}{40} $$

Then, multiply both sides by 40 to solve for \( \theta \):
\( \theta=\frac{5}{4}\times40 = 50 \)

Answer:

\( 50 \)