Question

in circle l, overarc{nop} is 90° and the radius is 5 units. which statement best describes the length of overarc{nop}?
\frac{1}{4} the area of circle l
\frac{1}{4} the circumference of circle l
\frac{1}{2} the area of circle l
\frac{1}{2} the circumference of

Explanation:

Step1: Recall the formula for arc length

The formula for the length of an arc \(s\) is \(s = r\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians). Also, the circumference \(C\) of a circle is \(C = 2\pi r\).

Step2: Convert the central angle to radians

Given the central angle \(\angle NLP=90^{\circ}\). Since \(180^{\circ}=\pi\) radians, then \(90^{\circ}=\frac{\pi}{2}\) radians.

Step3: Calculate the arc length \(s\) and compare with the circumference formula

Using \(s = r\theta\) with \(r = 5\) and \(\theta=\frac{\pi}{2}\), we have \(s=5\times\frac{\pi}{2}=\frac{5\pi}{2}\). The circumference \(C = 2\pi r=10\pi\). Now, \(\frac{s}{C}=\frac{\frac{5\pi}{2}}{10\pi}=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\) the circumference of circle \(L\)