Question

1. figure bcde and point o are shown.

a. draw and label the image of figure bcde under a dilation with center o and scale factor \\(\frac{1}{4}\\).
b. compare the distance between o and c to the distance between o and c.
c. based on the properties of dilations, what is true about \\(\overline{cd}\\) and \\(\overline{cd}\\)?

1. what is the length of \\(\overline{ab}\\)?

Explanation:

For Question 13:

a. To draw the dilated figure \(BCDE'\):

1. Draw lines from center \(O\) through each vertex \(B, C, D, E\).
2. On each line, mark a new point \(B', C', D', E'\) such that the distance from \(O\) to the new point is \(\frac{1}{4}\) the distance from \(O\) to the original vertex.
3. Connect \(B', C', D', E'\) to form the dilated quadrilateral.

b. Dilations scale all distances from the center by the scale factor. The distance \(OC'\) is \(\frac{1}{4}\) the length of \(OC\), or \(OC = 4 \times OC'\).

c. Dilations preserve the direction (parallelism) of line segments and scale their lengths by the scale factor. So \(\overline{CD} \parallel \overline{C'D'}\), and the length of \(\overline{C'D'}\) is \(\frac{1}{4}\) the length of \(\overline{CD}\).

Step1: Identify coordinates of A, B

Point \(A = (-4, 3)\), Point \(B = (2, -2)\)

Step2: Apply distance formula

The distance formula is \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
Substitute values:
\(d = \sqrt{(2 - (-4))^2 + (-2 - 3)^2}\)

Step3: Simplify the expression

\(d = \sqrt{(6)^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61}\)

Answer:

a. (Drawn figure: Quadrilateral \(B'C'D'E'\) smaller than \(BCDE\), centered around \(O\), with each vertex on the line from \(O\) to the original vertex, \(\frac{1}{4}\) the distance from \(O\) to the original vertex)
b. The distance between \(O\) and \(C'\) is \(\frac{1}{4}\) the distance between \(O\) and \(C\).
c. \(\overline{CD}\) is parallel to \(\overline{C'D'}\), and \(C'D' = \frac{1}{4}CD\).

---

For Question 14: