Question

1. here are two circles. their centers are a and f. (2 points)

a. what is the length of segment ad? how do you know?
b. on the first circle, what segment is a diameter? how long is it?

1. a wire that has a length of 88 cm is bent into a circle. what is the radius of the circle? (2 points)

Explanation:

Step1: Recall the property of radius in a circle

In a circle, all radii are equal. Since \(AC = 4\) cm and \(AD\) is also a radius of the same circle (with center \(A\)), by the definition of radius (a line - segment from the center of a circle to a point on the circle), we can say \(AD=AC\).

Step2: Identify the diameter of the first circle

A diameter of a circle is a line - segment that passes through the center of the circle and has its endpoints on the circle. In the first circle (with center \(A\)), if we consider two radii \(AC\) and \(AE\) (assuming \(E\) and \(C\) are on the circle and \(A\) is the center), the diameter \(d = 2r\). Given \(r = 4\) cm.

Step3: Use the formula for the circumference of a circle

The formula for the circumference of a circle is \(C = 2\pi r\) (where \(C\) is the circumference and \(r\) is the radius). We are given that \(C = 88\) cm. Substitute \(C = 88\) into the formula \(C=2\pi r\). Assuming \(\pi=\frac{22}{7}\), we have \(88 = 2\times\frac{22}{7}\times r\).
First, simplify the right - hand side: \(2\times\frac{22}{7}\times r=\frac{44}{7}r\).
Then, solve for \(r\) by cross - multiplying: \(r=\frac{88\times7}{44}\).

Answer:

a. The length of segment \(AD\) is \(4\) cm. We know this because \(AD\) is a radius of the circle with center \(A\), and all radii of a circle are equal. Given \(AC = 4\) cm (a radius), so \(AD = 4\) cm.
b. A diameter of the first circle could be \(CE\) (assuming \(C\) and \(E\) are endpoints). The length of the diameter \(d = 2r\). Since \(r = 4\) cm, \(d=8\) cm.
For the wire bent into a circle:

$$r=\frac{88\times7}{44}=14$$

The radius of the circle formed by the wire is \(14\) cm.