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## Integrated math 1

### Course: Integrated math 1>Unit 15

Lesson 3: Rotations

# Rotating shapes

Sal is given a triangle on the coordinate plane and the definition of a rotation about the origin, and he manually draws the image of that rotation.

## Want to join the conversation?

• Is there an easier way or special trick to memorize each angle's algebraic solution? •   The way that I remember it is that 90 degrees and 270 degrees are basically the opposite of each other. So, (-b, a) is for 90 degrees and (b, -a) is for 270. 180 degrees and 360 degrees are also opposites of each other. 180 degrees is (-a, -b) and 360 is (a, b). 360 degrees doesn't change since it is a full rotation or a full circle. Also this is for a counterclockwise rotation. If you want to do a clockwise rotation follow these formulas: 90 = (b, -a); 180 = (-a, -b); 270 = (-b, a); 360 = (a, b).
I hope this helps!

Edit:
I'm sorry about the confusion with my original message above. Here is the clearer version:

The "formula" for a rotation depends on the direction of the rotation.

Counterclockwise:
90 degrees: (-b, a) or (-y, x)
180 degrees: (-a, -b) or (-x, -y)
270 degrees: (b, -a) or (y, -x)
360 degrees: (a, b) or (x, y)

Clockwise:
90 degrees: (b, -a) or (y, -x)
180 degrees: (-a, -b) or (-x, -y)
270 degrees: (-b, a) or (-y, x)
360 degrees: (a, b) or (x, y)

Hope this clears things up. :)
• yea sal, i have to agree with everyone i love your videos, but i did get a little confused on here. There's a lot going on. • The way that I remember it is that 90 degrees and 270 degrees are basically the opposite of each other. So, (-b, a) is for 90 degrees and (b, -a) is for 270. 180 degrees and 360 degrees are also opposites of each other. 180 degrees is (-a, -b) and 360 is (a, b). 360 degrees doesn't change since it is a full rotation or a full circle. Also this is for a counterclockwise rotation. If you want to do a clockwise rotation follow these formulas: 90 = (b, -a); 180 = (-a, -b); 270 = (-b, a); 360 = (a, b).
• Does this technique work with all polygons or just triangles? •  I don't know about the technique in the video, but this technique works with all polygons for each degree of rotation:
Rotation of 90 degrees - translate points to (-b, a)
Rotation of 180 degrees - translate points to (-a, -b)
Rotation of 270 degrees - translate points to (b, -a)
Rotation of 360 degrees - translate points to (a, b) which is just staying at the initial shape

Hope this helps.
• I understand that negative degree rotation is counterclockwise and a positive degree rotation is clockwise. However, I do not understand where to place the points after rotations. Is there any way I can solve it algebraically? There was so much going on in the video that I got really confused. • While you got it backwards, positive is counterclockwise and negative is clockwise, there are rules for the basic 90 rotations given in the video, I assume they will be in rotations review.
For + 90 (counterclockwise) and - 270 (clockwise) (x,y) goes to (-y,x)
For + 180 or - 180 (the same) (x,y) goes to (-x,-y)
For + 270 or -90, (x,y) goes to (y,-x)
• Throughout the video, Sal uses the fact that a negative degree is clockwise and a positive degree is anti-clockwise. Why is this? • Imagine a line segment on a coordinate plane that rotates around the origin like a clock hand. If it travels 45° (or any positive degree angle), it will travel in a counterclockwise direction. If it rotates -45° degrees (or any negative degree angle), it will travel in a clockwise direction. The same thinking applies to any rotation. =)
• This has been really confusing for me but does rotating a shape just mean rotating each point of the shape? • Yes, you got it!

Each point is rotated about (or around) the same point - this point is called the point of rotation.

The key is to look at each point one at a time, and then be sure to rotate each point around the point of rotation.

Also, remember to rotate each point in the correct direction: either clockwise or counterclockwise.

In this video, you are told that the point of rotation is the origin (0, 0), but the point of rotation doesn't always have to be the origin.

Hope this helps! • Here's something that helps me visualize it:

So draw a shape/point on a paper. Put another paper on top of it (I like to imagine this one as being something like a transparent sheet protector, and I draw on it using a dry-erase marker) and trace the point/shape. Now place your finger on the rotation point. In this video's case, that's the origin. With your finger firmly on that point, rotate the paper on top. The shape is being rotated! But how do we do this for a specific angle?

Well, let's say the shape is a triangle with vertices A, B, and C, and we want to rotate it 90 degrees. The point at which we do the rotation, we'll call point P. The rotated triangle will be called triangle A'B'C'. As per the definition of rotation, the angles APA', BPB', and CPC', or the angle from a vertex to the point of rotation (where your finger is) to the transformed vertex, should be equal to 90 degrees. If you want, you can connect each vertex and rotated vertex to the origin to see if the angle is indeed 90 degrees.

I hope this gives you more of an intuitive sense.
• i'm going to have a mental breakdown   