Question

what is the scale factor of the dilation? options: \\(\frac{3}{4}\\), \\(\frac{1}{4}\\), 3, 4 (with a coordinate grid showing original figure with points e, f, g and dilated figure with points e, f, g)

Explanation:

Step1: Identify coordinates of pre - image and image points

Let's take point \( E \) and its image \( E' \). From the graph, assume the coordinates of \( E \) are \((-4, 4)\) and the coordinates of \( E' \) are \((-1, 1)\). (We can also use other corresponding points like \( F \) and \( F' \) or \( G \) and \( G' \))

Step2: Calculate the scale factor

The scale factor \( k \) of a dilation centered at the origin (since the lines from the origin pass through corresponding points) is given by the ratio of the coordinates of the image point to the pre - image point. For the \( x \) - coordinate: \( k=\frac{x_{E'}}{x_{E}}=\frac{- 1}{-4}=\frac{1}{4}\). For the \( y \) - coordinate: \( k = \frac{y_{E'}}{y_{E}}=\frac{1}{4}\). We can verify with another point, say \( F \) and \( F' \). If \( F=(4,4) \) and \( F'=(1,1) \), then \( k=\frac{1}{4}\) for both \( x \) and \( y \) coordinates.

Answer:

\(\frac{1}{4}\)