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Rotations in the x-y Plane


In the previous video, I talked about reflections in the x-y plane. Sometimes the test also asks about rotations in the x-y plane. In these cases, the center of rotation will almost always be the origin, and the angle will either be 90 degrees, one way or the other, or 180 degrees. First, think about quadrants. Any point that rotates 90 degrees clockwise will go down a quadrant.

If it's in quadrant IV, it would go to III, from III it will go to II, from II it will get to I, from I it will go to IV, that's the clockwise direction, IV to III to II to I, back to IV. Any point that rotates 90 degrees counterclockwise will go up a quadrant. So, if it's in I, it's gonna get moved to II, from II to III, from III to IV, and from IV back to I.

So I, II, III, IV back to I, that's the counterclockwise direction of rotation. Any point that rotates in 180 degrees will move to the opposite quadrant. So we swap back and forth between I and III, or swap back and forth between II and IV. Now we can be a little more precise. Recall the discussion about perpendicular slopes a few lessons ago.

For a 90 degree rotation either way, the x-distances, and y-distances swap place. So we see a triangle, the purple triangle has a horizontal leg, a long horizontal leg, and a short vertical leg. When we rotate it 90 degrees, anything horizontal becomes vertical, and anything vertical becomes horizontal. So the horizontal and vertical switch places in a 90 degree rotation.

So the new x value has the same absolute value as the old y value, and vice versa. We have to think through plus and minus signs for the new coordinates based on the quadrants. When we rotate by 180 degrees, the rule is even easier. If an original point, anywhere in the x-y plane is rotated 180 degrees, then the x- and y-coordinates of the new point have the same absolute value, and simply the opposite plus and minus signs.

So suppose we have these points here, think about what would happen if we rotated 180 degrees around the origin. Well, the positive 5 and positive 3 would just become both negatives. The 4, -1 would become -4, positive 1, and the two negatives would become 2 positive. And so those would be the three images under 180 degree rotation. Sometimes, we have to know what happens to the coordinates of individual points when we rotate.

In other questions, we just have to use our visual reasoning abilities. If this given shape is rotated by say, 180 degrees, what will be its new position and new orientation? So there's not really a formula for this, we have to use visual reasoning. If this is something you find challenging, practice. For example, draw, or maybe find in a magazine an asymmetrical figures, something that is clearly asymmetrical.

Look at it one way and then try to sketch or imagine what it would look like if it rotated 90 degrees clockwise or counterclockwise, or rotated 180 degrees. And then actually rotate the figure, and check and see how close you were. So do the sketch first, where you imagine it, and then actually rotate the figure and see how close you were. Here's a practice problem.

Pause the video, and then we'll talk about this. Okay, so that figure is in the first quadrant, we're gonna rotate it 180 degrees, that means it's gonna wind up in the third quadrant. So right away, we know that it's not gonna be in the second quadrant. And, if it rotates 180 degrees, it gets flipped over. So these two long sides right now, they're pointed down.

If it's flipped 180 degrees over, they're gonna be pointing up, and so the answer is D. The test will ask us to rotate things in the x-y plane, almost always by either 90 degrees, 90 degrees clockwise, 90 degrees counterclockwise, or 180 degrees, and almost always around the origin, so that's make it easier. The test could give us coordinates of any one point and asks to find the coordinate of the new rotated point.

Or, the test may simply give us a shape or figure on one quadrant and ask us to visualize how it would appear when rotated.

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