Geometry (all content)
An older video where Sal finds the images of shapes under various rotations, and where he determines the rotations that take one shape to another. Created by Sal Khan.
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- It would be helpful if you incorporated the algebraic steps to these results, i.e. how the x and y change for 90, 180 and 270 rotations. Problems often require this approach, and it's more useful, especially when the shape does not have a point on either axis (or there is no shape at all, i.e. just points given).(46 votes)
- When you rotate exactly 90 degrees around the origin, you will cross one axis. So take the (x,y) coordinates for the point, and see which axis would be crossed. In the rotated point, the x-value and y-value swap places. Then, whichever one doesn't match the axis that got crossed becomes negative.
Here's an example: "Rotate the point (4,1) around the origin by 90 degrees". The coordinates for the rotated point will be (-1,4). The 4 and the 1 swapped places, then the x-value became negative because the y-axis got crossed.
To rotate 180 or 270 degrees, you could keep rotating the point by 90 degrees until you arrive where you need to be. Or you could just memorize that rotating 180 degrees around the origin means both the x-value and the y-value become negative, without swapping places. So (2,3) rotated 180 degrees around the origin would be (-2,-3). Rotating 270 degrees is the same as rotating 90 degrees in the opposite direction.(24 votes)
- How do you find the rotation if it's not at the origin?(15 votes)
- Find the translation needed to move the center of rotation to the origin, apply that to the shape, perform your rotation, and then undo the original transformation(11 votes)
- what are the rules for rotation like if you go 90 degrees clockwise would you switch the coordinates and make one negative? I would like to know the rules for if you go 90 degrees counterclockwise, 90 degrees clockwise, 180 degrees counterclockwise, 180 degrees clockwise, 270 degrees counterclockwise, and finally 270 degrees clockwise. please help !!(8 votes)
- 90 clockwise and 270 counter clockwise are the same thing
90 counter clockwise and 270 clockwise are the same thing
180 clockwise and 180 counter clockwise are the same thing
90 counter clockwise would be (-y, x)
90 clockwise would be (y,-x)
180 either way would be (-x,-y)(11 votes)
- Is rotating by -90 the same as rotating by 270?
(all measures are in degrees)(3 votes)
- i still do not get how to solve this...are there any easy tricks for solving this?(3 votes)
- Why is the positive direction counterclockwise? This seems counterintuitive.(2 votes)
- Just look at a protractor and you will understand. The 0 degrees is on the right side and the numbers go counterclockwise.(3 votes)
- at0:31, it said you should go into counterclockwise direction. Why cant we go in clockwise direction?
Also, when you are dealing with shapes irregularly put in position, how would you know if it is 90 degrees?(2 votes)
- I don't understand how to get the tool to set at point 0,0 and then getting it to rotate counterclockwise to a measured degrees. How do you know you are at 90, 250, 300 degrees rotation, and how is this measured with this online tool?(2 votes)
- With a Question that goes "Rotate this shape 270 degrees" or something like that, Isn't a 270 degree turn identical to a 90 degree turn but in the other direction?(2 votes)
- That is technically correct. Good observation! :D It's kind of the same way with 180 degrees; you can turn a shape 180 degrees in either direction to have the shape end up in the same spot. But, turning the shape counterclockwise technically gives you a positive degree answer, while turning clockwise gives you a negative degree answer. If it's too confusing and you don't get it, comment this, OK? Hope this is helpful. ;D(1 vote)
What is the image of the polygon below after the rotation? So this is a rotational transformation of 270 degrees. So what we-- and when they talk about a rotational transformation, at least in the context of this exercise, we're talking about a rotation around the origin, so around this point. So the way I like to think about it is pick one of these points, one of these orange points, and, essentially, rotate it around the origin 270 degrees. So let's do that. So this would be rotating it. And when we talk about positive angles, we're talking about going in the counterclockwise directions. The same way that we, typically, measure angles when we're doing trigonometry with the unit circle and things like that. So here, so we started here, and let's rotate 90 degrees. Then we could keep rotating, rotate another 90 degrees so we've rotated a total of 180 degrees. And then we rotate 90 more degrees to get to 270. Let's do a couple more of these. What is the transformation that rotates the blue solid shape to the orange dashed shape? Now, this one's a little bit-- we don't have the manipulative here to help us answer that question so this is really going to challenge our powers of visualization. So the main thing is to pick a point on the blue shape and see where does it end up on the orange dashed shape. And I'll pick this point right over here because you can see it's pointing straight to the top left. And then when you go onto the orange dashed shape, it's pointing to the bottom right. So this one went all the way around. So you could say it went-- it was pointing from this direction and went all the way to going in the exact opposite direction. So this was a 180-degree transformation, a 180-degree rotation. Let's do a few more. So what's the image of the polygon? So they want us to do another 270-degree rotation. So like before. So let's rotate this 3/4 of the way around. So let's see how we can do that. So right now-- so this is going to be a little bit trickier. So if we-- actually, I'm going to pick this point because it's easier to visualize the 270 degrees. So now we've done 90 degrees. Now we can do 180 degrees. And now we've done 270 degrees.