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A geometric rotation refers to the rotating of a figure around a center of rotation. This lesson will get you going on rotations, give you some examples, and end with a quiz that tests your knowledge of rotations.

Geometric Transformations

There are four types of geometric transformations:

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  1. Translation – movement of the object without rotating it or changing its size
  2. Reflection – flipping the object about a line of reflection
  3. Rotation – rotating a figure about a point
  4. Dilation – changing the size of a figure without changing its essential shape

For this lesson, we’ll focus specifically on rotation.

Rotations

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A rotation is the movement of a geometric figure about a certain point. The amount of rotation is described in terms of degrees.

If the degrees are positive, the rotation is performed counterclockwise; if they are negative, the rotation is clockwise. The figure will not change size or shape, but, unlike a translation, will change direction. The initial figure is always called the pre-image, while the rotated figure will be called the image.Let’s look at an example. Let’s rotate this figure 90 degrees:

Rotate the figure 90 degrees
The figure was rotated 90 degrees
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Notation

The mathematical notation for rotation is usually written like this: R (center, rotation), where the center is the point of rotation and the rotation is given in degrees. Often, rotations are written using coordinate notation, which means that their coordinates on the coordinate plane are given. This will help you to draw both the pre-image and the image easily.

Rules of Rotation

There are some general rules for the rotation of objects using the most common degree measures (90 degrees, 180 degrees, and 270 degrees).

The general rule for rotation of an object 90 degrees is (x, y) ——–> (-y, x). You can use this rule to rotate a pre-image by taking the points of each vertex, translating them according to the rule, and drawing the image. Take the previous example: the points that mark the ends of the pre-image are (1, 1) and (3, 3).

When you rotate the image using the 90 degrees rule, the end points of the image will be (-1, 1) and (-3, 3).The rules for the other common degree rotations are:

  • For 180 degrees, the rule is (x, y) ——–> (-x, –y)
  • For 270 degrees, the rule is (x, y) ——–> (y, –x)

Returning to our first example, the end points of the image, if the pre-image were rotated 180 degrees, they would be (-1, -1) and (-3, -3); if it were rotated 270 degrees, the end points would be (1, -1) and (3, -3). Here is what all those rotations would look like on the graph:

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The rules for negative rotation are as follows:

  • -90 degrees, the rule is (x, y) ——–> (y, –x)
  • -180 degrees, the rule is (x, y) ——–> (-x, –y)
  • -270 degrees, the rule is (x, y) ——–> (-y, x)

Examples

‘Using notation, rotate the point (3, -5) about the origin 270 degrees.’Since the rule for a 270 degrees rotation is (x, y) ——–> (y, –x), the new point would be (-5, -3).

‘Rotate the following figure 180 degrees:’

Rotate the figure 180 degrees
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By using the rule for a 180 degrees rotation, we can get the coordinates for the image:

  • (2, 1) becomes (-2, -1)
  • (4, 1) becomes (-4, -1)
  • (2, 4) becomes (-2, -4)

So, the image after rotation is:

Figure rotated 180 degrees
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Lesson Summary

Rotation is a geometric transformation that involves rotating a figure a certain number of degrees about a fixed point. A positive rotation is counterclockwise and a negative rotation is clockwise.

You can use the rotational rules to determine where on the coordinate plane to place the vertices of your image, and then it is easy to draw the image from the pre-image.

Learning Outcomes

Once you are finished, you should be able to:

  • List the types of geometric transformations
  • Explain what a rotation is in math
  • Recall the rules of rotations
  • Rotate a geometric figure on a coordinate plan

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