The problem has two parts: 7. Calculate the arc length of two arcs given their radii and the central angle. a) Find the arc length of arc $CB$ if the radius is $7$ meters. (The central angle associated with $CB$ is $42^\circ$ from the diagram with arc $LN$.) b) Find the arc length of arc $EAB$ if the radius is $20$ millimeters. (The central angle associated with $EAB$ is $42^\circ$ from the diagram with arc $LN$.) 8. Find the measures of three angles in a circle. (The angles are $m\angle H$, $m\angle F$, and $m\angle G$.)

1. Problem Description

The problem has two parts:

7. Calculate the arc length of two arcs given their radii and the central angle.

a) Find the arc length of arc $CB$ if the radius is $7$ meters. (The central angle associated with $CB$ is $42^\circ$ from the diagram with arc $LN$.)

b) Find the arc length of arc $EAB$ if the radius is $20$ millimeters. (The central angle associated with $EAB$ is $42^\circ$ from the diagram with arc $LN$.)

8. Find the measures of three angles in a circle. (The angles are $m\angle H$, $m\angle F$, and $m\angle G$.)

2. Solution Steps

7. The formula for the arc length $s$ is given by $s = r \theta$, where $r$ is the radius and $\theta$ is the central angle in radians.

a) Given $r = 7$ meters and the central angle is $42^\circ$, we first convert the angle to radians: $\theta = 42^\circ \times \frac{\pi}{180^\circ} = \frac{42\pi}{180} = \frac{7\pi}{30}$ radians.

Then, the arc length $s = 7 \times \frac{7\pi}{30} = \frac{49\pi}{30} \approx 5.13127$ meters. Rounding to the nearest hundredth, we get $5.13$ meters.

b) Given $r = 20$ millimeters and the central angle is $42^\circ$, we first convert the angle to radians: $\theta = 42^\circ \times \frac{\pi}{180^\circ} = \frac{42\pi}{180} = \frac{7\pi}{30}$ radians.

Then, the arc length $s = 20 \times \frac{7\pi}{30} = \frac{140\pi}{30} = \frac{14\pi}{3} \approx 14.66076$ millimeters. Rounding to the nearest hundredth, we get $14.66$ millimeters.

8. From the given circle diagram, we have:

a) $\angle F$ is the central angle and $\angle H$ is an inscribed angle that intercepts the same arc as $\angle F$. Therefore, $m\angle H = \frac{1}{2} m\angle F$. Without further information on the measure of $\angle F$, we cannot calculate $m\angle H$. Looking at the diagram and assuming $\angle F$ intercepts $180^\circ$, then $m\angle F = 90^\circ$ and thus $m\angle H = 45^\circ$.

b) If $\angle F$ intercepts $180^\circ$, then $m\angle F = 90^\circ$.

c) $\angle G$ is the exterior angle and is congruent with $\angle F$, therefore $m\angle G = \angle F = 90^\circ$.

3. Final Answer

7.

a) $5.13$ meters

b) $14.66$ millimeters

8.

a) $45^\circ$

b) $90^\circ$

c) $90^\circ$