Question

in circle h with ( mangle ghj = 114^{circ} ) and ( gh = 13 ) units, find the length of ( overset{\frown}{gj} ). round to the nearest hundredth.

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.
First, convert the angle from degrees to radians. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). Given \(\theta = 114^{\circ}\), then \(\theta=\frac{114\pi}{180}=\frac{19\pi}{30}\) radians.
Since \(GH\) is the radius of the circle \(r = 13\) units.

Step2: Calculate the arc - length

Substitute \(r = 13\) and \(\theta=\frac{19\pi}{30}\) into the arc - length formula \(s=r\theta\).
\(s = 13\times\frac{19\pi}{30}=\frac{247\pi}{30}\)
Now, calculate the numerical value: \(\frac{247\pi}{30}\approx\frac{247\times3.14159}{30}\)
\(247\times3.14159 = 247\times(3 + 0.14159)=247\times3+247\times0.14159=741+34.97273 = 775.97273\)
\(\frac{775.97273}{30}\approx25.86576\)

Answer:

\(25.87\)