The problem asks us to find the length of arcs CB and EB given the radius and the central angle subtended by each arc. Also, we need to find measures of angles in a circle and a quadrilateral.

1. Problem Description

The problem asks us to find the length of arcs CB and EB given the radius and the central angle subtended by each arc. Also, we need to find measures of angles in a circle and a quadrilateral.

2. Solution Steps

7. a) We are asked to find the length of arc CB if the radius is 7 meters. The central angle $\angle COW$ is given as $57^{\circ}$. The length of an arc $s$ is given by the formula:

$s = r\theta$

where $r$ is the radius and $\theta$ is the central angle in radians. First, convert the central angle from degrees to radians:

$\theta = 57^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{57\pi}{180} = \frac{19\pi}{60}$ radians.

Then, the arc length $CB$ is

$s = 7 \times \frac{19\pi}{60} = \frac{133\pi}{60} \approx \frac{133 \times 3.14159}{60} \approx 6.9666$

Rounding to the nearest hundredth, we get 6.97 meters.

7. b) We are asked to find the length of arc EB if the radius is 20 millimeters. The central angle $\angle EOW$ is given as $58^{\circ}$. We follow the same procedure as in part (a). Convert the central angle from degrees to radians:

$\theta = 58^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{58\pi}{180} = \frac{29\pi}{90}$ radians.

Then, the arc length EB is

$s = 20 \times \frac{29\pi}{90} = \frac{58\pi}{9} \approx \frac{58 \times 3.14159}{9} \approx 20.2365$

Rounding to the nearest hundredth, we get 20.24 millimeters.

9. Given a cyclic quadrilateral ABCD, where $\angle B = 85^{\circ}$. In a cyclic quadrilateral, opposite angles are supplementary, which means they add up to $180^{\circ}$.

$\angle B + \angle D = 180^{\circ}$

$\angle D = 180^{\circ} - \angle B = 180^{\circ} - 85^{\circ} = 95^{\circ}$.

Also, $\angle A + \angle C = 180^{\circ}$.

We are given that $\angle A = 90^{\circ}$ and $\angle C = 90^{\circ}$.

So, $\angle A = 90^{\circ}$, $\angle B = 85^{\circ}$, $\angle C = 90^{\circ}$, $\angle D = 95^{\circ}$.

3. Final Answer

7. a) 6.97 meters

7. b) 20.24 millimeters

9. $\angle A = 90^{\circ}$

$\angle B = 85^{\circ}$

$\angle C = 90^{\circ}$

$\angle D = 95^{\circ}$