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## High school geometry

### Course: High school geometry>Unit 1

Lesson 6: Dilations

# Performing transformations FAQ

## What is a rigid transformation, and how is it different from a dilation?

A rigid transformation is a type of geometric transformation that doesn't change the size or shape of the object being transformed. Translations, rotations, and reflections are all rigid transformations.
Dilations are the one type of transformation in this unit that are not rigid. When you dilate an object, you change its size, but not its shape. Dilations can make objects bigger or smaller.

### How do we find the center of dilation?

First, we need to remember what dilation is: a geometric transformation that changes the size of an object without changing its shape.
The center of dilation, then, is the point from which the object is being scaled.
To find the center of dilation, we'll need to look at two corresponding points from the original object (the pre-image) and the new object (the image).
We'll label the points A and B for the pre-image and A, prime and B, prime for the image.
To find the center of dilation, we'll draw two lines: one from A to A, prime and one from B to B, prime.
The point where the two lines intersect is the center of dilation.
Try it yourself with our Dilations: center exercise.

### How do we find the center of rotation?

The center of rotation is the point around which we turn the points in the plane when we rotate a figure.
To find the center of rotation, we'll need to look at two corresponding points from the original object (the pre-image) and the new object (the image).
We'll label the points A and B for the pre-image and A, prime and B, prime for the image.
To find the center of rotation, we'll draw two segments: one from A to A, prime and one from B to B, prime.
Then we construct the perpendicular bisectors of start overline, A, A, prime, end overline and start overline, B, B, prime, end overline.
The point where the two perpendicular bisectors intersect is the center of rotation.

### How do we construct geometric transformations?

To translate a figure using a compass and straightedge, we'll need to measure the distance and direction we want to move the figure.
1. Start by drawing a vector (a line segment with an arrowhead) that shows the distance and direction we want to translate our figure.
2. Use the compass to measure the length of the vector.
3. Use the straightedge to draw a line parallel to the vector and starting at one of the points on our original figure.
4. Using the measurement we took with the compass, we can mark the point on the parallel line that is the correct distance away from the original figure.
5. We can repeat this process for each point on our original figure to create the translated figure.
To reflect a figure, we'll need to start with a line of reflection.
1. Draw the line of reflection.
2. Mark a point on the figure that we want to reflect.
3. Construct a perpendicular line to the line of reflection through that point.
4. Use the compass to measure the distance from the point to intersection of the perpendicular line and the line of reflection.
5. With the same compass setting, put the needle on the intersection point and make a mark where the pencil touches the perpendicular line on the opposite side of the line of reflection from the original figure.
6. Repeat this process for every point on the figure.
7. Use the straightedge to connect the points.
To rotate a figure, we'll need a center of rotation.
1. Mark a point as the center of rotation.
2. Use the compass to measure the distance from the point of rotation to one the vertices of the figure.
3. Draw an arc centered at the center of rotation through the vertex.
4. Use the protractor, centered on the center of rotation, to measure the angle. Go counterclockwise for positive angle measures and clockwise for negative angle measures. (Bonus: for certain angles, like 60, degree rotations, we can construct the angle instead of using a protractor.)
5. Use the straightedge to draw a line through the center of rotation along that angle. The point where the line intersects the arc is the corresponding vertex of the image.
6. Repeat this process for every point on the figure.
7. Use the straightedge to connect the points.

## Where do we see these types of transformations in the real world?

There are countless applications for geometric transformations in the real world! For example, architects might use translations, rotations, and reflections when designing a building, while graphic designers might use them to create logos or advertisements. Dilations are often used in map-making to scale a map up or down to the desired size.