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## High school geometry

# 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 $\stackrel{\u2015}{P{P}^{\prime}}$ .

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 $\mathrm{\u25b3}ABC$ to $\mathrm{\u25b3}{A}^{\prime}{B}^{\prime}{C}^{\prime}$ .

The center of rotation must be on the perpendicular bisector of $\stackrel{\u2015}{A{A}^{\prime}}$

The center of rotation also must be on the perpendicular bisector of $\stackrel{\u2015}{B{B}^{\prime}}$ .

We could also check the perpendicular bisector of $\stackrel{\u2015}{C{C}^{\prime}}$ , but we don't need to. Since all of the bisectors intersect at the same point, checking two is enough.

### Let's try it!

## Determining angle of rotation

Once we have found the center of rotation, we have several options for determining the angle of the rotation.

- Try to measure it with a protractor.
- Try to estimate it using benchmark angles.
- See how to calculate it using the law of cosines (once we've learned some trigonometry and have the coordinates).

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 $\mathrm{\u25b3}ABC$ to $\mathrm{\u25b3}{A}^{\prime}{B}^{\prime}{C}^{\prime}$ about point $P$ .

We can compare $m\mathrm{\angle}AP{A}^{\prime}$ to benchmark angles.

The angle measure is a little closer to $180\mathrm{\xb0}$ than to $90\mathrm{\xb0}$ . We could split the circle into more equal parts to get a closer estimate.

We might estimate that the angle is around $150\mathrm{\xb0}$ to $160\mathrm{\xb0}$ , 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 $-200\mathrm{\xb0}$ .

### Let's try it!

## Want to join the conversation?

- why are positive rotations counter clockwise?(86 votes)
- The reason for this actually comes from trigonometry. The representation of angles on the coordinate plane is called Standard Position. As regular angles are made up of two rays, so are angles in standard position. The rays in standard position angles, however, have specific names; the initial side, which always stays on the positive x-axis (to the right of the origin), and the terminal side, which rotates about the origin to create the angle. If the angle is positive, the terminal side rotates counter clockwise, and if the angle is negative, the terminal side rotates clockwise.

For example, if the terminal side was on the the positive y-axis (above the origin), then the angle made would be 90 degrees, because the terminal side rotated 90 degrees counter clockwise. Hope this helps!(135 votes)

- How will this help me out with daily life(44 votes)
- I'm learning to develop games in my free time, so learning geometry and trigonometry with this perspective makes way more sense to me now than on school days(10 votes)

- None of this makes any sense(27 votes)
- yep. thats math.(19 votes)

- Alright, I am writing this for guys from the future who will come for help here in the comments...So it is said that the center of rotation is always some point on the perpendicular bisector of the two points that we want to rotate.

But what exactly is the perpendicular bisector? It is a line that intersects a segment, splits it in two halves and creates a right angle. That is cool but why is it useful? It is useful because EVERY POINT ON THIS LINE (the perpendicular bisector)is at the same distance from the two end points of the segment. So the CENTER, CENTER, CENTER of rotation must be somewhere on this perpendicular bisector because IT'S THE CENTER of rotation meaning that it is equidistant from the starting point and the ending point. Cheers!(25 votes)- You are exactly right!(1 vote)

- I just want to go home(15 votes)
- I just want to go home too and see my family again but they're being held hostage by Sal till I finish rotations(13 votes)

- how would you go about doing this on graph paper(14 votes)
- You will need a protractor to help you figure out the angle and possibly a compass to help you accurately draw the rotated the figure. This video explains how to draw a rotated image: https://youtu.be/U4Hv494HwrQ(11 votes)

- I'm still a bit unsure as to how I am supposed to find the degree of rotation? The article is helpful but I don't feel like I should rely on the Khan Academy tool to figure out a rotation? Are there any videos/articles that explain how to do this without relying on the rotation tool? Sorry if this question is confusing!(14 votes)
- If you wanted to find the angle of rotation on graph paper you would:

1) Draw a line segment from one point on the original shape (let's call it Point A) to the center of rotation.

2) Draw a line segment from the image of Point A (Point A') to the center of rotation.

3) Measure the angle that they form using a protractor.

There's no formula to figure it out as far as I know of. You would either use a tool on a graphing calculator similar to the Khan Academy tool or the method I described above for solving on graph paper.(8 votes)

- Why are positive notations countered clockwise and why are negative notations clockwise?(14 votes)
- copying from another comment (I did not write this, @eiloncohen did):

The reason for this actually comes from trigonometry. The representation of angles on the coordinate plane is called Standard Position. As regular angles are made up of two rays, so are angles in standard position. The rays in standard position angles, however, have specific names; the initial side, which always stays on the positive x-axis (to the right of the origin), and the terminal side, which rotates about the origin to create the angle. If the angle is positive, the terminal side rotates counter clockwise, and if the angle is negative, the terminal side rotates clockwise.

For example, if the terminal side was on the the positive y-axis (above the origin), then the angle made would be 90 degrees, because the terminal side rotated 90 degrees counter clockwise. Hope this helps!(1 vote)

- how do you know a close estimate?(14 votes)
- confusing very confusing i think i might explode(11 votes)