Question

in circle g with m∠fgh = 54° and fg = 3 units, find the length of \\(\overarc{fh}\\). round to the nearest hundredth.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( \widehat{FH} \) in a circle is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. First, we need to convert the central angle from degrees to radians.
To convert degrees to radians, use the formula \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). Here, \( \theta = 54^\circ \), so \( \theta_{radians} = 54 \times \frac{\pi}{180} = \frac{3\pi}{10} \) radians.

Step2: Identify radius

The radius \( r \) of the circle is equal to \( FG \), which is 3 units (since \( FG \) is a radius of circle \( G \)).

Step3: Calculate arc length

Using the arc length formula \( s = r\theta \), substitute \( r = 3 \) and \( \theta = \frac{3\pi}{10} \):
\( s = 3 \times \frac{3\pi}{10} = \frac{9\pi}{10} \approx \frac{9\times 3.1416}{10} \approx 2.82744 \)

Step4: Round to nearest hundredth

Rounding \( 2.82744 \) to the nearest hundredth gives \( 2.83 \).

Answer:

\( 2.83 \)