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## Geometry (all content)

### Course: Geometry (all content)>Unit 10

Lesson 3: Rotations

# Rotating shapes: center ≠ (0,0)

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

## Want to join the conversation?

• Is there an easier mathematical way to do this, because this will take too long for tests.
• Okay, it took me a while to figure out a pattern, but there is an easier way to do by graphing. Create a pretend origin by drawing a dotted line Y-axis and X-axis where the arbitrary point is at. Then rotate your paper literally counter clockwise or clockwise whatever degrees you need it. You will see the dotted "pretend origin" has rotated. The shape in question also has rotated. Now again draw another "pretend orirgin2" at the arbitrary point as if it didn't move, You will see the "shift" or translation between the first pretend origin and the second. Remember that shape in question? Translate that shape using the same "shift" between the first and second pretend origin. Saves you TIME!
• Here's my idea about doing this in a bit more 'mathematical' way: every rotation I've seen until now (in the '...about arbitrary point' exercise) has been of a multiple of 90°. Because the axes of the Cartesian plane are themselves at right angles, the coordinates of the image points are easily predictable: with a bit of experimentation, you could easily 'prove' to yourself that rotating (a, b) 90° would result in (-b, a) [by the way, this has some interesting consequences in trig]; rotation of 180° gives us (-a, -b) and one of 270° would bring us to (b, -a). Rotating (a, b) 360° would result in the same (a, b), of course.

Now, since (a, b) are coordinates with respect to the origin, this only works if we rotate around that point. But it's easy to calibrate it if you want to specify another point, around which you want to rotate -- just make that point the new origin! Figure out the other points' coordinates with respect to your new origin [if the new origin had coordinates (x, y), then (a, b)'s new coordinates would be (a-x, b-y) ], do the transformations, and then translate everything back to coordinates with respect to the old origin.
• Yeah, I was thinking about that too (that is, doing this in a more mathematical way). Computer languages employ this shifting of the matrix to simplify transformations for the programmer, but I don't know if that is how the calculations are actually done "under the hood".
• Is there some kind of formula for this, because I want to take less time and I'm pretty sure there must be a formula, Please?
• There is actually a mathematical way to do this, to find the explanation, you can go to the top questions section, and the first question and answer should be it, the answer was provided by thaao hanshew.
• How are we supposed to get the question in Rotations 1 & 2 correct when we don't even have the scratchpad available. Sal has a program that he uses to write with ( black background), but we don't have that. I have to wing the questions and just hope for the best. How have you guys been doing the questions?
• It might help to just get out a piece of paper and a pencil when the scratchpad is not available. Most of the time, for harder problems, I prefer paper over the scratchpad anyways.

Otherwise, you could always get a basic drawing program like MS paint (or a drawing app if you're on a tablet) and use that in place of the scratchpad.
• Can we rotate it by using the patterns of the rotations and thinking of the point of rotation as the origin? I think if we redraw the Y-axis and X-axis with the point and used the patterns it would work.
• Yes it would work actually, but the mathematical way is easier.
• Is there a formula that we can use for arbitrary points? it is hard to do the exercise after this video if one lacks a lot of visualization skills.
• Wouldn't it be easier to just translate the second and third point? For example, once you have rotated a point (e.g M to M'), why not just figure out the translation of M to M' and translate the other points accordingly? This is a rigid body so distances and angles will be preserved.
(1 vote)
• If you rotate around something (whether something close like a baseball bat for a dizzy race, or far like a hissing cobra you want to avoid but can't get far enough away from), you change not only your position but the way you're facing. It's different from if you move across the room without changing the way you face. If you merely translate the figure, you risk neglecting the change in "direction" that comes with a rotation.
• I'm a bit confused. If the initial image is 270° rotated, and we want to draw the image before rotation, I think we have to rotate the image by more 90° or -270°. This will take our image to its initial position.
In this video Sal rotate the image by -90° that is the same rotation already applied to the image and ends up with an image that is rotated by -180° or 180° that is the same. Does someone note this too?
(1 vote)
• The question might be worded a bit ambiguously, but it means that you are required to rotate the initial image 270*. It says:
SAM is rotated 270*. Draw the image of this rotation on the interactive graph.
Therefore, SAM has already been rotated the set amount (here 270) of degrees around a certain point. We need to draw the image of that pre-done rotation on the graph.
I hope this helps you! Have a great day!
• Can we rotate it by using the patterns of the rotations and thinking of the point of rotation as the origin? I think if we redraw the Y-axis and X-axis with the point and used the patterns it would work.
• That is exactly what you want to do. Shift the origin to the center of rotation.
(1 vote)
• How do you find the center of rotation when you have the original figure and the rotated figure when it is not the origin?