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## High school geometry

# 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?(57 votes)
- 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. :)(256 votes)

- 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.(77 votes)
- 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).(0 votes)

- For all people who don't understand, here is a much simpler way to do this:

90° counterclockwise: (x,y) -> (-y,x)

180°: (x, y) -> (-x,-y)

270° counterclockwise rotation: (x,y) -> (y,-x)

for example, if you have a point with coordinates (2,8) and you want to rotate it 90 degrees counterclockwise, it becomes (-8,2)

I hope that helps :)(59 votes)- Thank youuuuu(3 votes)

- Does this technique work with all polygons or just triangles?(11 votes)
- 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.(31 votes)

- 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.(16 votes)
- 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)(17 votes)

- Throughout the video, Sal uses the fact that a negative degree is clockwise and a positive degree is anti-clockwise. Why is this?(14 votes)
- 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. =)(17 votes)

- I learned this in a much simpler way. The key is with every point, when its being rotated by 90", you switch from (x,y) to (y,x), and add in the psitive or negative sign according to which direction and plane it will be in. For example, if point A is being rotated 90" to the right, and its current position is (1,3), then its new position is going to be (3,-1). I hope that makes sense.(27 votes)

- why is he making this so complicated?(12 votes)
- This video is more complicated than it needs to be. In order to rotate shapes, you just rotate all the points and then connect the dots, in the process giving you a triangle.(6 votes)

- This has been really confusing for me but does rotating a shape just mean rotating each point of the shape?(7 votes)
- 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!(12 votes)

- Someone please help, I really don't understand this at all!(8 votes)
- 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.(6 votes)

## Video transcript

- [Voiceover] We're told that triangle PIN is rotated negative 270
degrees about the origin. So this is the triangle PIN
and we're gonna rotate it negative 270 degrees about the origin. Draw the image of this rotation
using the interactive graph. The direction of rotation
by a positive angle is counter-clockwise. So positive is counter-clockwise, which is a standard convention, and this is negative, so a negative degree would be clockwise. And we want to use this tool here. And this tool, I can put points in, or I could delete points. I can draw a point by clicking on it. So what we want to do is think about, well look, if we rotate
the points of this triangle around the origin by negative 270 degrees, where is it gonna put these points? And to help us think about that, I have copied and pasted
this on our scratch pad. So actually let me go over here so I can actually draw on it. So let's just first think
about what a negative 270 degree rotation actually is. So if I were to start, if I were to, let me draw some coordinate axes here. So that's an x-axis and that's a y-axis. If you were to start right over here and you were to rotate around
the origin by negative, so this is the origin here, by negative 270 degrees, what would that be? Well let's see, this would
be rotating negative 90. This would be rotating
another negative 90, which would, together, be negative 180. And then this would be
another negative 90, which would give you in total, negative 270 degrees. That's negative 270 degrees. Now notice, that would
get that point here, which we could have also gotten there by just rotating it by
positive 90 degrees. We could have just said
that this is equivalent to a positive 90 degree rotation. So if they want us to
rotate the points here around the origin by negative 270 degrees, that's equivalent to just rotating all of the points, and I'll
just focus on the vertices, because those are the
easiest ones to think about, to visualize. I can just rotate each of
those around the origin by positive 90 degrees. But how do we do that? And to do that, what I am going to do, to do that what I'm gonna do is I'm gonna draw a series
of right triangles. So let's first focus on, actually let's first focus
on point I right over here. And let me draw right triangle. And I could draw it several ways. But let me draw it like this. So, it's a right triangle where the line between the origin and I is its hypotenuse. So let me see if I can, It's I could probably draw, actually I can use a line tool for that. So let me... So that's the hypotenuse of the line. Now if I'm gonna rotate I 90 degrees about the origin, that's
equivalent to rotating this right triangle 90 degrees. So what's going to happen there? Well this side, right over here, if I rotate this 90 degrees, where is that going to go? Well instead of going
seven along the x-axis, it's going to go seven along the y-axis. So it's going to, if you
rotate it positive 90 degrees, that side it going to look like this. So that's rotating it 90 degrees, just like that. Now what about this side over here? What about this side? Let me do this in a different color. What about, what about this side right over here? Well this side over here, notice we've gone down from the origin, we've gone down seven. But if you were to rotate it up, notice this forms a right angle between this magenta
side and this blue side, so you're gonna form a right angle again. And so from this point,
you're gonna go straight as a right angle and
instead of going down seven, you're going to go to the right seven. So you're gonna go to the right seven, just like this. And so the point I, or
the corresponding point in the image after the rotation is going to be right over here. So that green line, let me
draw the hypotenuse now, it's gonna look like, oops, I wanted to do that in a different color. I want to do that in the green. I have trouble changing colors. Alright, so there you go. It's gonna look like that. So your new point I, if you rotate this triangle,
this right triangle 90 degrees, your new point I, maybe I shouldn't say, I'll call it I prime, which is what is the image of this point after I've done the 90 degree rotation is going to be right over here. And now we can do that
for each of the points. We can do that for N here. Let's draw a right triangle. Let's draw a right triangle. And I can do it a bunch of different ways. I could draw a right triangle like this. Let me draw it like this. So that's one side of my right triangle. Actually, let me draw
the hypotenuse first. So I have the hypotenuse. Connects the origin to
my point, just like that. And then I could either draw it up here, or I could draw it down here, like this. I could draw it like this. So if you rotated this 90 degrees, if you rotate this 90 degrees, this side, which is seven units long and we're going seven below the origin, it's now going to be seven to the right. Right? If I rotate it by 90 degrees, it's going to be right over here. It's gonna be here and this side, which has length, let me switch colors. This side, which has length two, it forms a right angle, so we're gonna forma a right angle and have length two right over here. And so the image of
point N is going to be, let me get the right color. It is going to be just like that. It is going to be the image of point N is going to be right here. I'll call that N prime. So we know where the new, where the image of I
is, the image of N is, so now we just have to
think about where does the image of P sit? And once again, we can do our
little right triangle idea. So let's draw a right triangle. So, just like that. I could draw that side. And I can do this side, right over here. So if I were to rotate, if I were to focus on this, let me do this in a color I haven't. So if I were to focus on, I've already used that color. If I were to focus on
this, right over here, and if I were to rotate it by 90 degrees. Instead of going two to the right, it's going to go two straight up. If I rotate this by 90 degrees
it's gonna be just like this. Now, this side on, let me pick another color. This side, right over
here, forms a right angle and it has a length of three. So we're gonna forma right angle here and have a length of three. And just like that, we know where the image of P is going to be. It is going to be, it is going to be right over here. So this is P prime. And I know it's kind of confusing, but now we just have to connect the P prime, the I prime, and the N prime to figure out what the
image of my triangle is after rotation. So let me do that. So if I connect these two, I get that. If I connect these two, I get that. And if I connect these two, if I connect those two, I have that. And there you have it, I have the image. And now I just have to input it on the actual problem. So, let's see, point
negative three comma two, let me get it out. So negative three comma two is there. I have seven comma two, so let me put that in there. Seven comma two. And then I have seven
comma seven is there. So let me get this out. So seven comma seven. And then I'll draw here
again to connect the lines. And I'm done. I've rotated it through
an angle of 90 degrees. Or negative 270 degrees, which is what they originally asked me for. And I can check to make
sure I got the right answer.