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Lesson Explainer: Rotations on the Coordinate Plane Mathematics • 8th Grade

In this explainer, we will learn how to find the vertices of a shape after it undergoes a rotation of 90, 180, or 270 degrees about the origin clockwise and counterclockwise.

Let us start by rotating a point. Recall that a rotation by a positive degree value is defined to be in the counterclockwise direction.

Take the point (3,4), which is located in the top-right part of the π‘₯𝑦-plane (i.e., the first quadrant). We will call this point 𝐴.

Rotating point 𝐴 by 90 degrees about the origin gives us point 𝐴′ at coordinates (βˆ’4,3). This is made clearer by connecting line segments from the origin to points 𝐴 and 𝐴′, from which we can see that a right angle is formed.

Notice the reoccurrence of the 3 and 4 from the coordinates of point 𝐴. In fact, all rotations about the origin in multiples of 90 degrees will follow similar patterns. In general terms, rotating a point with coordinates (π‘₯,𝑦) by 90 degrees about the origin will result in a point with coordinates (βˆ’π‘¦,π‘₯).

Now, consider the point (3,4) when rotated by other multiples of 90 degrees, such as 180, 270, and 360 degrees. We will add points 𝐴′′ and 𝐴′′′ to our diagram, which represent point 𝐴 rotated by 180 and 270 degrees counterclockwise respectively. Notice that rotating point 𝐴 by 360 degrees will bring it back to where it started, to the coordinates (3,4).

The rotation of a point (π‘₯,𝑦) by 180 degrees is represented by the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,βˆ’π‘¦). The rotation of a point (π‘₯,𝑦) by 270 degrees is represented by the coordinate transformation (π‘₯,𝑦)β†’(𝑦,βˆ’π‘₯). The rotation of a point (π‘₯,𝑦) by 360 degrees does not alter its coordinates, and such a rotation can be represented by the coordinate transformation (π‘₯,𝑦)β†’(π‘₯,𝑦).

We also note that all rotations about the same point that differ by a multiple of 360 degrees are equivalent. This is because rotating by 360 degrees brings us back to where we started. For example, rotating by 380∘ is the same as rotating by 360∘ and then by 20∘; since the rotation of 360∘ does not change the coordinates, rotating by 380∘ is the same as rotating by 20∘ only. We must also note that these rotations are only equivalent when they are about the same point. Rotations around different points can never be equivalent unless the rotation is zero degrees (or equivalent to a rotation of zero degrees).

Furthermore, a negative (clockwise) rotation can always be reexpressed as a positive (counterclockwise) rotation. As an example, we will find a positive equivalent to a rotation of βˆ’90∘. Since rotations that differ by a multiple of 360∘ are equivalent, we can add 360∘ to the βˆ’90∘ rotation, which gives us 270∘ in total. Hence, a rotation of βˆ’90∘ is equivalent to a rotation of 270∘, and, therefore, both rotations can be expressed as the coordinate transformation (π‘₯,𝑦)β†’(𝑦,βˆ’π‘₯).

Property: Rotations by Multiples of 90 Degrees about the Origin

For a point with coordinates (π‘₯,𝑦), the following is true:

  • A rotation of 90 degrees results in a point with coordinates (βˆ’π‘¦,π‘₯).
  • A rotation of 180 degrees results in a point with coordinates (βˆ’π‘₯,βˆ’π‘¦).
  • A rotation of 270 degrees results in a point with coordinates (𝑦,βˆ’π‘₯).
  • A rotation of 360 degrees results in a point with coordinates (π‘₯,𝑦).
  • A rotation with a positive degree value indicates a counterclockwise rotation, and a rotation with a negative degree value indicates a clockwise rotation.
  • A rotation of βˆ’90∘ is equivalent to a rotation of 270∘ and therefore results in a point with coordinates (𝑦,βˆ’π‘₯).
  • A rotation of βˆ’180∘ is equivalent to a rotation of 180∘ and therefore results in a point with coordinates (βˆ’π‘₯,βˆ’π‘¦).
  • A rotation of βˆ’270∘ is equivalent to a rotation of 90∘ and therefore results in a point with coordinates (βˆ’π‘¦,π‘₯).
  • A rotation of βˆ’360∘ is equivalent to a rotation of 0∘ and therefore results in a point with coordinates (π‘₯,𝑦).

Now that we have introduced rotations of points, we will discuss rotating line segments and polygons on the coordinate plane. Since line segments and polygons can be defined by points, specifically the endpoints of a line segment or the vertices of a shape, rotating these is a matter of applying the coordinate transformations to multiple points.

For example, consider triangle 𝑃𝑄𝑅 with vertices 𝑃(2,1), 𝑄(3,5), and 𝑅(6,4). To rotate this triangle 180 degrees about the origin, we need to rotate each of its points according to the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,βˆ’π‘¦). The rotated triangle 𝑃′𝑄′𝑅′ will have vertices 𝑃′(βˆ’2,βˆ’1), 𝑄′(βˆ’3,βˆ’5), and 𝑅′(βˆ’6,βˆ’4).

When verifying whether an image is the correct rotation of a preimage, we can apply the coordinate transformations on the endpoints or vertices of the preimage and see if they match the coordinates on the new image.

For example, if we are given the graph above and asked to verify that triangle 𝑃′𝑄′𝑅′ is a 180∘ rotation of triangle 𝑃𝑄𝑅 about the origin, we will first take point 𝑃(2,1). Applying the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,βˆ’π‘¦) gives us (βˆ’2,βˆ’1), confirming that these are the coordinates of point 𝑃′. After we confirm this for points 𝑄(3,5) and 𝑄′(βˆ’3,βˆ’5) and for points 𝑅(6,4) and 𝑅′(βˆ’6,βˆ’4), we can verify that triangle 𝑃′𝑄′𝑅′ is indeed the image of triangle 𝑃𝑄𝑅 after a 180∘ rotation about the origin.

Lastly, we note that all rotations share a special property. Because rotating a figure will not change its absolute size or shape, rotations are called a rigid transformation.

Definition: Rigid Transformation

A rigid transformation of a figure is a transformation that preserves the distance between each pair of points in the figure.

A rigid transformation is sometimes referred to as an isometry.

For example, if the distance from point 𝑃 to point 𝑄 is equal to √10 and they are rotated about the same point 𝑅, then the distance between the new points 𝑃′ and 𝑄′ will still be √10. Rotations, reflections, and translations are examples of rigid transformations.

Now, we will work through some example problems involving rotations on the coordinate plane. First, we will take a look at an example of using the coordinate transformation of a rotation to rotate a shape.

Example 1: Using the Coordinate Transformation of a Rotation to Rotate a Shape

What is the image of 𝐴𝐡𝐢𝐷 under the transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯)?

Answer

Let us apply the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯) to each point in 𝐴𝐡𝐢𝐷. In each case, we want to swap the π‘₯- and 𝑦-coordinates then reverse the sign of the first coordinate. Doing this gives us 𝐴(4,3)βŸΆπ΄β€²(βˆ’3,4),𝐡(5,3)βŸΆπ΅β€²(βˆ’3,5),𝐢(5,4)βŸΆπΆβ€²(βˆ’4,5),𝐷(4,5)βŸΆπ·β€²(βˆ’5,4).

Therefore, the image of 𝐴𝐡𝐢𝐷 under the transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯) is given by the points 𝐴′(βˆ’3,4), 𝐡′(βˆ’3,5), 𝐢′(βˆ’4,5), and 𝐷′(βˆ’5,4).

The rotation appears to be a 90-degree counterclockwise rotation. This matches our knowledge that such a rotation is represented by the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), which is the coordinate transformation applied in the problem.

Next, we will look at an example of identifying equivalent rotations when given a certain angle of rotation.

Example 2: Identifying Equivalent Rotations

Which of the following is equivalent to a 25∘ rotation about the origin?

  1. A βˆ’335∘ rotation about the origin
  2. A βˆ’25∘ rotation about the origin
  3. A βˆ’155∘ rotation about the origin
  4. A 25∘ rotation about the point (1,1)
  5. A 335∘ rotation about the origin

Answer

We may disregard option D as it is a rotation around a different point. Rotations around different points will never be equivalent unless the rotation is zero degrees or an angle that is equivalent to zero degrees.

Rotations about the same point are equivalent when they differ by a multiple of 360 degrees. Thus, to determine the correct answer, we can add or subtract multiples of 360 degrees until we arrive at one of the other answers.

While adding 360 degrees does not give us any of the options shown, if we subtract 360 degrees, we get 25βˆ’360=βˆ’335.∘∘∘

Therefore, 25 degrees and βˆ’335 degrees are equivalent rotations. Hence, option A is correct.

Next, we will look at an example of identifying the image of a shape after a rotation about the origin.

Example 3: Identifying the Image of a Shape after a Rotation about the Origin

If triangle 𝐴𝐡𝐢 is rotated by 𝑅(𝑂,90)∘, which triangle would represent its final position?

Answer

𝑅(𝑂,90)∘ indicates a rotation of 90 degrees counterclockwise about the origin. Visually, it is possible to see that a shape in the top-right part of the graph (i.e., the first quadrant) would move to the top-left part (i.e., the second quadrant) after a 90-degree counterclockwise rotation, and so the correct answer is either option B or E, which have figures in the second quadrant. From there, we can see that option B is a reflection in the 𝑦-axis rather than a rotation. Option E is correctly rotated 90 degrees counterclockwise from the original figure.

We can also prove this mathematically, as this rotation can also be expressed through the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯).

Take point 𝐴 from the preimage. It is located at (1,1). After the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), point 𝐴′ should be located at (βˆ’1,1). Thus, we can narrow our choices down to option B or option E, as those are the only choices with point 𝐴′ at (βˆ’1,1).

Next, consider point 𝐡 from the preimage. It is located at (3,1). After the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), point 𝐡′ should be located at (βˆ’1,3). Between options B and E, only option E has point 𝐡′ at (βˆ’1,3).

We may also consider point 𝐢 from the preimage, which is located at (1,2). After the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), point 𝐢′ should be located at (βˆ’2,1). Option E does indeed show 𝐢′ at this location.

Hence, the correct answer is option E, as this is the only choice where points 𝐴′, 𝐡′, and 𝐢′ all match the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), representing a 90∘ counterclockwise rotation about the origin.

Next, we will look at an example where we must determine the vertices of a triangle rotated about the origin, this time without accompanying diagrams in the choices.

Example 4: Rotating a Triangle about the Origin

Determine the coordinates of the vertices’ images of triangle 𝐴𝐡𝐢 after a counterclockwise rotation of 270∘ around the origin.

Answer

A counterclockwise rotation of 270 degrees about the origin, which can be notated as 𝑅(𝑂,270)∘, can be represented by the coordinate transformation (π‘₯,𝑦)β†’(𝑦,βˆ’π‘₯). To find the image of the shape after the rotation, we can apply this transformation to each of its vertices in turn.

Applying this coordinate transformation to point 𝐴, the coordinates of which are (βˆ’7,βˆ’3), gives us 𝐴′ at (βˆ’3,7).

Applying this coordinate transformation to point 𝐡, the coordinates of which are (βˆ’3,βˆ’4), gives us 𝐡′ at (βˆ’4,3).

Applying this coordinate transformation to point 𝐢, the coordinates of which are (βˆ’6,βˆ’5), gives us 𝐢′ at (βˆ’5,6).

To check our work, we can visualize where these points would fall on the graph, and we can see that our new points lie in the second quadrant. This matches our prior knowledge that a 270-degree rotation about the origin would move a shape from the third quadrant to the second quadrant.

Hence, the coordinates of the vertices’ images of triangle 𝐴𝐡𝐢 after a counterclockwise rotation of 270∘ around the origin are 𝐴′(βˆ’3,7), 𝐡′(βˆ’4,3), and 𝐢′(βˆ’5,6).

Lastly, we will look at an example where we must apply our knowledge of the properties of rotation as a rigid transformation.

Example 5: Understanding the Properties of Rotation

In the figure, 𝐴𝐡 has been rotated 90∘ counterclockwise about the origin. Is the length of the image 𝐴′𝐡′ resulting from this transformation greater than, less than, or the same as the length of 𝐴𝐡?

Answer

Rotations are a β€œrigid transformation,” which means that distances between points are preserved through the transformation. Since the lengths of these line segments are exactly the distances between their two endpoints and the distance between these two points is preserved through the transformation, the length of the image 𝐴′𝐡′ is the same as the length of 𝐴𝐡.

Let us finish by recapping some key points from the explainer.

Key Points

  • A rotation of π‘₯ degrees is equivalent to a rotation of (π‘₯Β±360) degrees.
  • Coordinate transformations can be used to find the images of rotated points as follows:
    • A rotation of 90 degrees counterclockwise about the origin is equivalent to the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯).
    • A rotation of 180 degrees counterclockwise about the origin is equivalent to the coordinate transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,βˆ’π‘¦).
    • A rotation of 270 degrees counterclockwise about the origin is equivalent to the coordinate transformation (π‘₯,𝑦)β†’(𝑦,βˆ’π‘₯).
    • A rotation of 360 degrees about the origin is equivalent to a rotation of 0 degrees and both are equivalent to the coordinate transformation (π‘₯,𝑦)β†’(π‘₯,𝑦).
  • Line segments and shapes can be rotated by applying coordinate transformations to each of their endpoints or vertices.
  • To confirm if an image on a coordinate plane is a rotation of a given preimage, we can apply the appropriate coordinate transformation to each point from the preimage and then verify if each point matches the corresponding point in the image.
  • Rotations are rigid transformations, which means that distances are preserved through the transformation.

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