Question

dilate by a factor of 1/3
dilate by a factor of 2/3
pre - image image
pre - image image
pre - image image
pre - image image
pre - image image
pre - image image
scale factor:
scale factor:
scale factor:
scale factor:

Explanation:

Step1: Recall dilation rule

To dilate a figure by a scale - factor \(k\), we multiply the coordinates of each vertex of the pre - image by \(k\).

Step2: Analyze first dilation (top - left)

The scale factor is \(\frac{1}{3}\). If we assume a vertex of the pre - image has coordinates \((x,y)\), the coordinates of the corresponding vertex of the image are \((\frac{1}{3}x,\frac{1}{3}y)\).

Step3: Analyze second dilation (top - right)

The scale factor is \(\frac{2}{3}\). For a vertex \((x,y)\) of the pre - image, the vertex of the image has coordinates \((\frac{2}{3}x,\frac{2}{3}y)\).

Step4: Analyze third dilation (middle - left)

Since the pre - image and image are shown with the pre - image as a dotted figure and image as a solid figure, and the image is smaller. By comparing the lengths of corresponding sides, we can see that the scale factor is \(\frac{1}{2}\) (for example, if we measure the length of a side of the pre - image and the corresponding side of the image).

Step5: Analyze fourth dilation (middle - right)

The scale factor is \(\frac{1}{3}\) as the image is smaller than the pre - image and by comparing the side - lengths of the pre - image and image, we find that the ratio of the side - length of the image to the side - length of the pre - image is \(\frac{1}{3}\).

Step6: Analyze fifth dilation (bottom - left)

The scale factor is \(\frac{1}{2}\) as the image is smaller than the pre - image and by comparing corresponding side - lengths, we can determine this ratio.

Step7: Analyze sixth dilation (bottom - right)

The scale factor is \(\frac{1}{4}\) as the image is smaller than the pre - image and by comparing the lengths of corresponding sides, we find that the ratio of the side - length of the image to the side - length of the pre - image is \(\frac{1}{4}\).

Answer:

Top - left: \(\frac{1}{3}\)
Top - right: \(\frac{2}{3}\)
Middle - left: \(\frac{1}{2}\)
Middle - right: \(\frac{1}{3}\)
Bottom - left: \(\frac{1}{2}\)
Bottom - right: \(\frac{1}{4}\)