Question

1. after a dilation centered at the origin, the image of \\(\overline{cd}\\) is \\(\overline{cd}\\). if the coordinates of the endpoints of the segment are \\(c(6, -4)\\); \\(d(10, -8)\\); \\(c(27, -18)\\); and \\(d(45, -36)\\), what is the scale factor of the dilation and is it an enlargement or reduction? show your work!!!

Explanation:

Step1: Recall dilation formula

For a dilation centered at the origin, the coordinates of the image \((x', y')\) of a point \((x, y)\) are given by \(x' = kx\) and \(y' = ky\), where \(k\) is the scale factor. So, we can find \(k\) by dividing the \(x\)-coordinate (or \(y\)-coordinate) of the image point by the \(x\)-coordinate (or \(y\)-coordinate) of the original point.

Step2: Calculate scale factor using x - coordinates of C and C'

For point \(C(6, - 4)\) and its image \(C'(27, - 18)\), we use the formula \(k=\frac{x'}{x}\). Substituting \(x = 6\) and \(x'=27\) into the formula, we get \(k=\frac{27}{6}=\frac{9}{2} = 4.5\). We can also verify using the \(y\)-coordinates: \(\frac{- 18}{-4}=\frac{9}{2}=4.5\). Let's check with point \(D(10,-8)\) and \(D'(45, - 36)\). Using the \(x\)-coordinates: \(\frac{45}{10}=\frac{9}{2} = 4.5\), and using the \(y\)-coordinates: \(\frac{-36}{-8}=\frac{9}{2}=4.5\).

Step3: Determine if it's enlargement or reduction

Since the scale factor \(k = \frac{9}{2}=4.5>1\), when the scale factor of a dilation is greater than 1, the dilation is an enlargement.

Answer:

The scale factor of the dilation is \(\frac{9}{2}\) (or \(4.5\)) and it is an enlargement.