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## High school geometry

### Course: High school geometry>Unit 1

Lesson 4: Rotations

# Determining rotations

Learn how to determine which rotation brings one given shape to another given shape.
There are two properties of every rotation—the center and the angle.

## Determining the center of rotation

Rotations preserve distance, so the center of rotation must be equidistant from point $P$ and its image ${P}^{\prime }$. That means the center of rotation must be on the perpendicular bisector of $\stackrel{―}{P{P}^{\prime }}$.
If we took the segments that connected each point of the image to the corresponding point in the pre-image, the center of rotation is at the intersection of the perpendicular bisectors of all of those segments.

### Example

Let's find the center of rotation that maps $\mathrm{△}ABC$ to $\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$.
The center of rotation must be on the perpendicular bisector of $\stackrel{―}{A{A}^{\prime }}$
The center of rotation also must be on the perpendicular bisector of $\stackrel{―}{B{B}^{\prime }}$.
We could also check the perpendicular bisector of $\stackrel{―}{C{C}^{\prime }}$, but we don't need to. Since all of the bisectors intersect at the same point, checking two is enough.

### Let's try it!

Problem 1.1
$\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$ is the image of $\mathrm{△}ABC$ after a rotation.
Which point is the center of rotation?

## Determining angle of rotation

Once we have found the center of rotation, we have several options for determining the angle of the rotation.
Finally, we need to determine whether the rotation is counterclockwise, with a positive angle of rotation, or clockwise, with a negative angle of rotation.

### Example

Let's estimate the angle of rotation that maps $\mathrm{△}ABC$ to $\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$ about point $P$.
We can compare $m\mathrm{\angle }AP{A}^{\prime }$ to benchmark angles.
The angle measure is a little closer to $180\mathrm{°}$ than to $90\mathrm{°}$. We could split the circle into more equal parts to get a closer estimate.
We might estimate that the angle is around $150\mathrm{°}$ to $160\mathrm{°}$, but we'd have to measure to be sure.
We also could have measured clockwise, but then we would need to use a negative angle measure. We go a little more than a half turn clockwise, so we could estimate the angle measure to be around $-200\mathrm{°}$.

### Let's try it!

Problem 2.1
Triangle $\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$ is the image of $\mathrm{△}ABC$ under a rotation about point $P$.
What is the best estimate of the angle of rotation?