Question

polygon ABCD coordinates are: A (-1,-1), B (-1,1), C(2,3), D(2,-1)
polygon A'B'C'D' coordinates are: A'(-2,-2) B'(-2,2) C'(4,6) D'(4,-2)
In the image, polygon ABCD is dilated to form polygon A'B'C'D' with respect to the origin. What is the magnitude of dilation?

Answer 1

To find the magnitude of the dilation from polygon ABCD to polygon A'B'C'D', we can use the coordinates of any corresponding points to determine the scale factor.

1. Let's take point A and its image A':
- A: (-1, -1)
- A': (-2, -2)

The dilation scale factor, \( k \), can be calculated using the formula:

\[
k = \frac{A'}{A}
\]

This is calculated for both the x and y coordinates:

For the x-coordinate:
\[
k_x = \frac{x_{A'}}{x_A} = \frac{-2}{-1} = 2
\]

For the y-coordinate:
\[
k_y = \frac{y_{A'}}{y_A} = \frac{-2}{-1} = 2
\]

Since both scale factors \( k_x \) and \( k_y \) are equal, the magnitude of the dilation is the same for both axes.

We can double-check this with another corresponding point. Let’s use point B and its image B':
- B: (-1, 1)
- B': (-2, 2)

Calculate the dilation scale factor again:

For the x-coordinate:
\[
k_x = \frac{-2}{-1} = 2
\]

For the y-coordinate:
\[
k_y = \frac{2}{1} = 2
\]

Since we obtain the same scale factor, we can conclude that the magnitude of dilation is consistently found to be:

\[
\text{Magnitude of dilation} = 2
\]

Thus, the magnitude of dilation from polygon ABCD to polygon A'B'C'D' is \( \boxed{2} \).