The scale factor is the ratio between corresponding measurements of a figure and a representation of that figure.
D
What happens to the shadow on the wall?
A
Enlargement
B
Center of dilation
C
4/3
D
The shadow and the scale factor become larger. See solution.
Practice makes perfect
We know that we can use a flashlight and a shadow puppet to project shadows on the wall. Consider the following example.
As we can see, the shadow cast on the wall is larger than our hands. This means that the shadow is the image of our hands after a dilation. The type of dilation is an enlargement because it is larger than the preimage, which in our case is our hands.
Let's draw lines to connect corresponding vertices of the image and preimage. We can extend these lines to the flashlight.
Now, remember that if we draw lines connecting the corresponding vertices of the image and preimage, the lines will meet at a point. This point is called center of dilation. In our case, the light rays meet at the flashlight. Therefore, the flashlight is our center of dilation.
Consider a rabbit-shaped puppet shadow. The ears on the shadow puppet are 3 inches long. We want to find the scale factor of the dilation if the ears of the shadow are 4 inches long. To do so, remember that the scale factor is the ratio between corresponding measurements of a figure and a representation of that figure.
Scale Factor= Image Measurement/Preimage Measurement
In our case, the measurement of the image, the shadow, is 4 inches, and the measurement of the preimage, the puppet, is 3 inches.
Scale Factor= 4/3
Therefore, the scale factor is 43.
If we move the shadow puppet closer to the flashlight, we will see a larger shadow on the wall. This happens because our hands occupy more space in the light, which means that the size of the shadow will increase. Let's see an example!
Therefore, the scale factor of the dilation also increases.
