Question

question 5
what is the ratio of the side lengths on polygon fghji to the corresponding side lengths on polygon abcde? enter the ratio in the table. how does this ratio relate to the scale factor of the dilation?
side lengths
fg : ab
gh : bc
hi : cd
ij : de
jf : ea
ratio
space used(includes formatting): 1041 / 30000

Explanation:

To determine the ratio of side lengths of polygon \( FGHJI \) to \( ABCDE \) (and its relation to the dilation scale factor), we follow these steps:

Step 1: Recall Dilation Properties

Dilation is a transformation that scales a figure by a scale factor \( k \). For corresponding sides of the pre - image ( \( ABCDE \)) and image ( \( FGHJI \)), the ratio of the length of a side in the image to the length of the corresponding side in the pre - image is equal to the scale factor of the dilation.

Let the length of a side in \( ABCDE \) be \( L_{pre} \) and the length of the corresponding side in \( FGHJI \) be \( L_{post} \). The ratio we need is \( \frac{L_{post}}{L_{pre}} \), and this ratio is equal to the scale factor of the dilation.

Step 2: Assume (or Measure) Side Lengths (if not given numerically)

Since the problem does not provide specific numerical values for the side lengths, we rely on the property of dilation. In a dilation, all corresponding side - length ratios are equal and equal to the scale factor.

For example, if we consider side \( FG \) in \( FGHJI \) and side \( AB \) in \( ABCDE \), the ratio \( \frac{FG}{AB} \) is the same as \( \frac{GH}{BC} \), \( \frac{HI}{CD} \), \( \frac{IJ}{DE} \), and \( \frac{JF}{EA} \), and this ratio is the scale factor of the dilation.

If we assume (for a general case) that the scale factor of dilation from \( ABCDE \) to \( FGHJI \) is \( k \), then the ratio of the side lengths ( \( FGHJI:ABCDE \)) is \( k \). And this ratio is equal to the scale factor of the dilation.

Answer:

The ratio of the side lengths of polygon \( FGHJI \) to \( ABCDE \) is equal to the scale factor of the dilation. If we let the scale factor be \( k \), the ratio is \( k \) (and the ratio of side lengths is equal to the scale factor of dilation).

(Note: If specific side - length values were given, we would calculate the ratio as \( \frac{\text{Length of side in } FGHJI}{\text{Length of corresponding side in } ABCDE} \). For example, if \( AB = 2 \) and \( FG=4 \), the ratio would be \( \frac{4}{2}=2 \), and the scale factor of dilation would be 2.)