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Dilations - Concept
Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
A dilation is a non-rigid transformation, which means that the original and the image are not congruent. They are, however, similar figures. To perform dilations, a scale factor and a center of dilation are needed. If the scale factor is larger than 1, the image is larger than the original; if the scale factor is less than 1, the image is smaller than the original.
There are 4 types of transformations. Only one of these is not an isometry and that is a dilation. So a reflection, a rotation and a translation are all isometries. Which means when you perform the transformation, you're creating congruent figures. When you dilate a figure, you are creating similar figures which means the angles are going to be constant and the sides will be proportional to to each other.
When we dilate we need a scale factor, we'll call that n. If n is greater than 1 then your new image will be larger than the original. If n is less than 1 then your new image will be smaller than your original. To do this you need a center of dilation. Let's look at two examples to see why center of dilation is important.
When you go to the doctor's office and you have your pupils dilated, the center of dilation is the center of your pupil. So it starts off smaller, and it grows larger. So we're going to say that your scale factor here must have been larger than 1.
If you have a point in space and you are given a center of rotation, you're going to find your new point. How? By measuring this line segment between your center of dilation and your point and multiplying it times your scale factor. Here we can tell that your scale factor must be greater than 1. Why is that? Because our new point our prime is further away from our center of dilation than the original.
Let's say we had n less than 1. If we had our center of dilation here and some point r, if n is less than 1, we know that our new point r prime will somewhere on this line segment but it will be closer to your center of dilation.
The reason why center of dilation is important is because if I didn't give it to you, I could have drawn r prime over here, I could have drawn r prime basically anywhere in this plane and I would have dilated according to my scale factor.
So again, when you are dilating, you need to know your scale factor and a center of dilation. If your scale factor is larger than 1, it will grow larger. If it's less than 1, you will have a reduction.
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