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# Determining rotations

CCSS.Math:

Find the angle of a rotation by considering the source and the image of the rotation.

## Want to join the conversation?

- is there any way to find out how many degrees to turn it without guess and check?(26 votes)
- Well, I guess you can do it by looking at the coordinates and calculating it, but it's too complicated to explain and not worth doing. Since they give you an actual model of it when rotating, just give it a rough estimate and plug it in. That's all I can say. :)(20 votes)

- why is he saying prime? when he says a letter(11 votes)
- In the video:

ΔA'B'C' is the image of ΔABC under a rotation about the origin, (0, 0).

The**source**, ΔABC, is read "triangle A B C"

-*this is the triangle you start with*

The**image**, ΔA'B'C', is read "triangle A-prime B-prime C-prime"

-*this is the triangle you get after the rotation*

Using the suffix "prime" after each point lets us know that he is talking about the**image**of the rotation (not the**source**of the rotation).

Hope this helps!(19 votes)

- How do i tell if the rotation is negative is posative or negative?(9 votes)
- By convention, counter-clockwise rotation is positive, clockwise rotation is negative.(10 votes)

- But how do we know which way the shape was rotated? It could be any way. Sometimes the two shapes are really far apart and its really hard to tell.(5 votes)
- When you have practiced this enough, you should be able to tell the 4 general rotations (90 degrees, 180 degrees, and 270 degrees) counterclockwise (positive direction), and thus their equivalents (270 degrees, 180 degrees, and 90 degrees) clockwise. Whit this, you can at least be able to figure out a lot of limitations. so looking at the picture in the video, you should be able to see that it is < 90 counterclockwise (between 0 to 90) and which would be >270 clockwise (between - 270 and - 360 degrees). While it may not always give the answer, it generally eliminates 2 of 4 answers.(3 votes)

- Is there a reason it is called a prime?(4 votes)
- Yes, there is a reason it is called a prime!

Using the suffix "prime" after each point of ΔA'B'C' - "A-prime B-prime C-prime" - lets us know that we are talking about the*image*of the rotation, and not the*source*of the rotation (ΔABC, the triangle we started with).

Hope this helps!(4 votes)

- I don't get rotations can somebody help me(1 vote)
- What exactly don't you understand about them?(7 votes)

- At0:48, Sal mentions that counterclockwise means a positive angle, why is that?(5 votes)
- It has to do with the unit circle. A zero angle is the positive x axis. Rotating counterclockwise insures that all of the basic angles (0 < x < 90) are in the I (positive, positive) quadrant, and then other related angles in the II, III, and IVth quadrants are then easy to calculate.(0 votes)

- Could you make a video with some formulas?(3 votes)
- The formulas require sufficient knowledge of trigonometry & such. Rotation is used all the time in programming to make objects rotate, and is often combo-ed with linear algebra to make the cool effects you'll see in games & animations.

Khan Academy collaborated with Pixar to make this series of videos, where they teach you about deriving that formula: https://www.khanacademy.org/computing/pixar/sets/rotation/a/rotation-lesson-brief(2 votes)

- Why does the
**positive**angle turn`Counterclockwise`

and a protractor measured as`Clockwise`

?

Why We Can't Just change the positive angle to "`clockwise`

" that we will not**mixed up together**?

Please try to answer this because this gives me a**headache**.(3 votes)- It has to do with the unit circle and the trig functions. Since the 1st quadrant is positive x and positive y, it makes sense that the basic trig functions are within this quadrant. So the angles between 0 and 90 are in the first quadrant and require the counterclockwise rotation. Clockwise would put the angles in the positive x negative y fourth quadrant.(1 vote)

- How is it a positive angel?(2 votes)
- We define rotations in the counter-clockwise direction as positive and those going clockwise as negative. This is just an established convention, and if you wanted to do it the other way, you would have to restructure a fair amount of math.(3 votes)

## Video transcript

- [Instructor] We're told that triangle A-prime, B-prime, C-prime, so that's this red triangle over here, is the image of triangle ABC, so that's this blue triangle here, under rotation about the origin, so we're rotating about the origin here. Determine the angle of rotation. So like always, pause this video, see if you can figure it out. So I'm just gonna think about how did each of these
points have to be rotated to go from A to A-prime, or B to B-prime, or from C to C-prime? So let's just start with A. So this is where A starts. Remember we're rotating about the origin. That's why I'm drawing this
line from the origin to A. And where does it get rotated to? Well, it gets rotated to right over here. So the rotation is going in
the counterclockwise direction, so it's going to have a positive angle. So we can rule out these
two right over here. And the key question is, is
this 30 degrees or 60 degrees? And there's a bunch of ways
that you could think about it. One, 60 degrees would
be 2/3 of a right angle, while 30 degrees would
be 1/3 of a right angle. A right angle would look
something like this. So this looks much more
like 2/3 of a right angle, so I'll go with 60 degrees. Another way to think
about is that 60 degrees is 1/3 of 180 degrees, which this also looks
like right over here. And if you do that with any of the points, you would see a similar thing. So just looking at A to
A-prime makes me feel good that this was a 60-degree rotation. Let's do another example. So we are told quadrilateral A-prime, B-prime, C-prime,
D-prime, in red here, is the image of quadrilateral
ABCD, in blue here, under rotation about point Q. Determine the angle of rotation. So once again, pause this video, and see if you can figure it out. Well, I'm gonna tackle this the same way. I don't have a coordinate plane here, but it's the same notion. I can take some initial point
and then look at its image and think about, well, how
much did I have to rotate it? I could do B to B-prime, although this might be
a little bit too close. So I'm going from B to,
let me do a new color here, just 'cause this color is
too close to, I'll use black, so we're going from B to
B-prime right over here. We are going clockwise, so it's going to be a negative rotation. So we can rule that and that out. And it looks like a right angle. This looks like a right angle, so I feel good about
picking negative 90 degrees. We could try another
point and feel good that that also meets that negative 90 degrees. Let's say D to D-prime. So this is where D is initially. This is where D is, and this is where D-prime is. And once again, we are moving clockwise, so it's a negative rotation. And this looks like a right angle, definitely more like a right
angle than a 60-degree angle. And so this would be negative 90 degrees, definitely feel good about that.