Show that any dilation transforms any figure into a similar figure.
In the hours that have passed since you first asked this question, you could have spent 30 seconds with google and found what you needed. For instance,
https://learnzillion.com/lesson_plans/8580-prove-two-figures-are-similar-after-a-dilation
To show that any dilation transforms any figure into a similar figure, we need to prove that the resulting figure has the same shape as the original figure, but possibly a different size.
A dilation is a transformation that scales the size of an object by a certain factor. It is performed by multiplying the coordinates of each point in the figure by the scaling factor. The scaling factor can be greater than 1 (enlargement) or between 0 and 1 (reduction).
Let's take a look at the key points and angles in a figure before and after a dilation.
1. Key Points:
- Before dilation: Any key point in the original figure will be mapped to a new point in the dilated figure by multiplying its coordinates by the scaling factor.
- After dilation: The corresponding point in the dilated figure will have coordinates that are a scaled version of the corresponding point in the original figure.
2. Angles:
- Before dilation: The angles between any two line segments are maintained in the original figure.
- After dilation: The angles between the corresponding line segments in the dilated figure will be equal to the corresponding angles in the original figure.
From these observations, we can conclude that all corresponding angles in the original and dilated figures are equal and all corresponding sides in the original and dilated figures are proportional to each other.
Therefore, any dilation transforms any figure into a similar figure, as the shape of the figure is maintained while only the size is changed.