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

Lesson 9: Old transformations videos

# Determining rotations (old)

An older video where Sal determines the rotation that takes one shape to another. Created by Sal Khan.

## Want to join the conversation?

• So there is no method for this, it's just hit and mis?
• Since a rotation leaves the distance from the center unchanged, the center of rotation must be along the perpendicular bisector of the point and its image. Give two such point/image pairs, you can find the center of rotation.
• what rules would be performed if you had nothing but a pencil and paper?
• To find the point of rotation, find the point where point A is the same distance of the rotated point A. Then use this chart to help:
90° counterclockwise/270° clockwise
(x, y) → (-y, x)

180° either direction
(x, y) → (-x, -y)

270° counterclockwise/90° clockwise
(x, y) → (y, -x)
• Is there any rule to be able to rotate a random shape around a specific point?
• 1. Translate the shape and the center point so the new center point lands at the origin. So if the center point is (xc, yc) translate the shape by (-xc, -yc).
2. Rotation of point (x, y) by a specific angle a is done using the following equations, which yield the new point (xr, yr):
xr = x sin a + y cos a
yr = x cos a - y sin a
For rotations of multiples of 90 degrees these equations become simple but in general you need the trigonometric functions.
3. Translate the results back to the original axes, by (xc, yc).

So:
Xnew=(Xold-xc) sin(a) + (Yold-yc) cos(a) + xc
Ynew=(Xold-xc) cos(a) - (Yold-yc) sin(a) + yc
• No rule? Is it just a guess-and-check type of work, or is there a more specific process
• There are rules, such as for rotating (x, y) around the origin:
Rotating 90 degrees changes it to (-y, x)
Rotating 180 degrees changes it to (-x, -y)
Rotating 270 degrees changes it to (y, -x)
• How do you know the degree of rotation?
• Let a1 be the slope of the line segment connecting one original point to the center of rotation. Let a2 be the similar slope for the rotated point.

The angle of the line segment to the original point is arctan(a1), the angle of the line segment to the rotated point is arctan(a2), so the rotation angle is arctan(a2)-arctan(a1).
(1 vote)
• I've been through this transformations segment a couple of times now, but i need to know how to draw a reflection of a line segment over another line then over another one at a different angle please help
(1 vote)
• Have you tried sketching your problem out on graph paper? One thing you might try is to plot your segment and then fold the paper on the first line of reflection to see where it lands. Do the same thing with the second line of reflection. See where that takes you. Sometimes just using paper and pencil helps to focus our learning processes.
• How do you perform a rotation?
(1 vote)
• A rotation is when you turn a point, line, or shape on a coordinate plane. Sal gives an example in this video.
• Is there a video that shows how to rotate a shape around a point without the rotation tool?
(1 vote)
• what are the rules of rotation?
(1 vote)
• The rules of rotation are:
1) There must be a point around which the entire figure rotates.
2) If the points of the figure do not keep the same distance from the point around which the figure rotates, then you made a mistake and moved the entire figure or individual points away from the the point around which the figure rotates.
(1 vote)
• How do you find the rule for rotating triangles? This is the question and I don't understand it. How do you do it? Figure ABCD is a rectangle with point A (-2, 5). What rule would rotate the figure 90° clockwise, and what coordinate would be the output for point A'?
(1 vote)

## Video transcript

- [Instructor] Use one rotation to map quadrilateral ABCD to the other quadrilateral. So to map this one to this one right over here. Use a number between zero and 360 degrees to describe the angle. Counter-clockwise is positive. So we're going to want to move is counter-clockwise to try and get it to map there. And the only option they give us 'cause they want us to do it with one rotation is the rotation tool. And so we have to think about what do you we want to rotate around what point and if we put it right over here, it looks like this point point A does correspond to this point right over here. So if we were to rotate this around, if we were to rotate this around not 90 but it looks like 180 degrees around this point, point A would show up over here, it feels like point let's see is that right is that right or let's actually just try it out. Point A would show up over, no no no that's not right, that doesn't seem to... Let's try it out. 'Cause if we rotated 180 degrees, oh actually I was right it did match up, that's why this is interesting, it tests your visualization skills. So it did actually match up and what I did is I put that point of rotation exactly between those points but because it looked like 180 degrees around this point, so rotation by 180 degrees about negative one negative one so the center of rotation is negative one negative one, the angle of rotation is 180 degrees. Point A maps to this point right over here, so point A maps to the point one negative one, one negative one. And point C point C which is diagonally opposite point A maps to this point right over here, which is six negative six, six negative six, six negative six and we got it right.