In circle P with $$ m \angle N P Q=96^{\circ} $$ and $$ N P=19 $$ units, find the length of $$ \overparen{N Q} $$. Round to the nearest hundredth.

Question

UpStudy AI · Accepted Answer

1. Since $$ P $$ is the center of the circle, the radius is given by
   $$
   NP = 19 \text{ units.}
   $$
   
2. The central angle $$ \angle NPQ $$ measures $$ 96^\circ $$. To compute the arc length, we first convert the central angle to radians:
   $$
   96^\circ = 96 \times \left(\frac{\pi}{180}\right) = \frac{96\pi}{180} = \frac{8\pi}{15} \text{ radians.}
   $$
   
3. The formula for the length $$ s $$ of an arc corresponding to a central angle $$ \theta $$ (in radians) is:
   $$
   s = r \theta.
   $$
   Substituting the known values:
   $$
   s = 19 \times \frac{8\pi}{15} = \frac{152\pi}{15}.
   $$
   
4. Converting to a decimal value:
   $$
   s \approx \frac{152 \times 3.1416}{15} \approx \frac{477.522}{15} \approx 31.80.
   $$
   
Thus, the length of $$ \overparen{NQ} $$ is approximately $$ 31.80 $$ units.
The length of $$ \overparen{NQ} $$ is approximately 31.80 units.

In circle P with \( m \angle N P Q=96^{\circ} \) and \( N P=19 \) units, find the length of \( \overparen{N Q} \). Round to the nearest hundredth.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Geometry Questions