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What happens to a point in the coordinate plane when it is rotated 270 degrees about the origin?

Written By Wednesday, April 6, 2022 Add Comment Edit

In this explainer, we will larn 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 starting time past rotating a point. Retrieve that a rotation by a positive degree value is defined to be in the counterclockwise direction.

Accept the signal ( 3 , 4 ) , which is located in the top-correct part of the 𝑥 𝑦 -airplane (i.east., the first quadrant). We will call this betoken 𝐴 .

Rotating signal 𝐴 past 90 degrees virtually the origin gives united states of america point 𝐴 ′ at coordinates ( − 4 , 3 ) . This is made clearer past connecting line segments from the origin to points 𝐴 and 𝐴 ′ , from which nosotros tin encounter that a right bending is formed.

Notice the reoccurrence of the three and 4 from the coordinates of point 𝐴 . In fact, all rotations about the origin in multiples of 90 degrees volition follow similar patterns. In general terms, rotating a signal with coordinates ( 𝑥 , 𝑦 ) past ninety degrees about the origin will result in a point with coordinates ( − 𝑦 , 𝑥 ) .

Now, consider the point ( three , 4 ) when rotated by other multiples of ninety degrees, such as 180, 270, and 360 degrees. We will add points 𝐴 ′ ′ and 𝐴 ′ ′ ′ to our diagram, which represent point 𝐴 rotated past 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 ( three , 4 ) .

The rotation of a point ( 𝑥 , 𝑦 ) past 180 degrees is represented past the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑥 , − 𝑦 ) . The rotation of a signal ( 𝑥 , 𝑦 ) past 270 degrees is represented by the coordinate transformation ( 𝑥 , 𝑦 ) → ( 𝑦 , − 𝑥 ) . The rotation of a bespeak ( 𝑥 , 𝑦 ) by 360 degrees does not alter its coordinates, and such a rotation can exist represented by the coordinate transformation ( 𝑥 , 𝑦 ) → ( 𝑥 , 𝑦 ) .

Nosotros as well note that all rotations about the same indicate that differ by a multiple of 360 degrees are equivalent. This is because rotating past 360 degrees brings us dorsum to where we started. For case, rotating past 3 8 0 ∘ is the same as rotating by three vi 0 ∘ and so by 2 0 ∘ ; since the rotation of 3 6 0 ∘ does non modify the coordinates, rotating past 3 8 0 ∘ is the same as rotating by 2 0 ∘ only. Nosotros must likewise annotation that these rotations are only equivalent when they are about the aforementioned point. Rotations around different points can never be equivalent unless the rotation is z e r o degrees (or equivalent to a rotation of z e r o degrees).

Furthermore, a negative (clockwise) rotation can always be reexpressed as a positive (counterclockwise) rotation. Equally an example, we will detect a positive equivalent to a rotation of − 9 0 ∘ . Since rotations that differ by a multiple of 3 six 0 ∘ are equivalent, we tin add three half-dozen 0 ∘ to the − 9 0 ∘ rotation, which gives u.s. ii seven 0 ∘ in total. Hence, a rotation of − 9 0 ∘ is equivalent to a rotation of 2 7 0 ∘ , and, therefore, both rotations tin can be expressed as the coordinate transformation ( 𝑥 , 𝑦 ) → ( 𝑦 , − 𝑥 ) .

Property: Rotations by Multiples of 90 Degrees about the Origin

For a point with coordinates ( 𝑥 , 𝑦 ) , the post-obit is true:

  • A rotation of 90 degrees results in a point with coordinates ( − 𝑦 , 𝑥 ) .
  • A rotation of 180 degrees results in a indicate with coordinates ( − 𝑥 , − 𝑦 ) .
  • A rotation of 270 degrees results in a point with coordinates ( 𝑦 , − 𝑥 ) .
  • A rotation of 360 degrees results in a point with coordinates ( 𝑥 , 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 − 9 0 ∘ is equivalent to a rotation of 2 7 0 ∘ and therefore results in a point with coordinates ( 𝑦 , − 𝑥 ) .
  • A rotation of − 1 viii 0 ∘ is equivalent to a rotation of 1 8 0 ∘ and therefore results in a bespeak with coordinates ( − 𝑥 , − 𝑦 ) .
  • A rotation of − 2 vii 0 ∘ is equivalent to a rotation of 9 0 ∘ and therefore results in a point with coordinates ( − 𝑦 , 𝑥 ) .
  • A rotation of − three 6 0 ∘ is equivalent to a rotation of 0 ∘ and therefore results in a bespeak with coordinates ( 𝑥 , 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 tin can exist defined past 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 case, consider triangle 𝑃 𝑄 𝑅 with vertices 𝑃 ( 2 , 1 ) , 𝑄 ( three , 5 ) , and 𝑅 ( 6 , four ) . To rotate this triangle 180 degrees nigh the origin, we need to rotate each of its points according to the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑥 , − 𝑦 ) . The rotated triangle 𝑃 ′ 𝑄 ′ 𝑅 ′ will have vertices 𝑃 ′ ( − 2 , − 1 ) , 𝑄 ′ ( − 3 , − 5 ) , and 𝑅 ′ ( − 6 , − 4 ) .

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

For case, if we are given the graph above and asked to verify that triangle 𝑃 ′ 𝑄 ′ 𝑅 ′ is a one 8 0 ∘ rotation of triangle 𝑃 𝑄 𝑅 well-nigh the origin, we will first take point 𝑃 ( two , 1 ) . Applying the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑥 , − 𝑦 ) gives the states ( − ii , − 1 ) , confirming that these are the coordinates of point 𝑃 ′ . After nosotros confirm this for points 𝑄 ( 3 , v ) and 𝑄 ′ ( − 3 , − 5 ) and for points 𝑅 ( six , 4 ) and 𝑅 ′ ( − half-dozen , − 4 ) , we tin can verify that triangle 𝑃 ′ 𝑄 ′ 𝑅 ′ is indeed the image of triangle 𝑃 𝑄 𝑅 afterwards a one eight 0 ∘ rotation about the origin.

Lastly, nosotros note that all rotations share a special belongings. Because rotating a effigy volition not alter 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 every bit an isometry.

For instance, if the distance from point 𝑃 to signal 𝑄 is equal to √ 1 0 and they are rotated about the same point 𝑅 , then the distance between the new points 𝑃 ′ and 𝑄 ′ will still be √ one 0 . Rotations, reflections, and translations are examples of rigid transformations.

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

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

What is the epitome of 𝐴 𝐵 𝐶 𝐷 under the transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) ?

Reply

Permit united states of america employ the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) to each point in 𝐴 𝐵 𝐶 𝐷 . In each example, we desire to swap the 𝑥 - and 𝑦 -coordinates then contrary the sign of the first coordinate. Doing this gives u.s.a. 𝐴 ( four , 3 ) ⟶ 𝐴 ′ ( − 3 , 4 ) , 𝐵 ( 5 , 3 ) ⟶ 𝐵 ′ ( − 3 , 5 ) , 𝐶 ( 5 , 4 ) ⟶ 𝐶 ′ ( − iv , 5 ) , 𝐷 ( four , 5 ) ⟶ 𝐷 ′ ( − v , 4 ) .

Therefore, the image of 𝐴 𝐵 𝐶 𝐷 nether the transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) is given past the points 𝐴 ′ ( − 3 , 4 ) , 𝐵 ′ ( − 3 , 5 ) , 𝐶 ′ ( − 4 , 5 ) , and 𝐷 ′ ( − 5 , 4 ) .

The rotation appears to be a ninety-degree counterclockwise rotation. This matches our knowledge that such a rotation is represented by the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) , which is the coordinate transformation practical in the trouble.

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

Example 2: Identifying Equivalent Rotations

Which of the following is equivalent to a 2 5 ∘ rotation about the origin?

  1. A − 3 3 5 ∘ rotation nearly the origin
  2. A − 2 5 ∘ rotation almost the origin
  3. A − 1 v 5 ∘ rotation about the origin
  4. A ii 5 ∘ rotation about the betoken ( one , one )
  5. A 3 3 five ∘ rotation nearly the origin

Answer

Nosotros may disregard pick D as it is a rotation around a different point. Rotations around different points will never be equivalent unless the rotation is z east r o degrees or an angle that is equivalent to z e r o degrees.

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

While adding 360 degrees does non requite us whatsoever of the options shown, if nosotros subtract 360 degrees, we get ii 5 − iii six 0 = − iii iii 5 . ∘ ∘ ∘

Therefore, 25 degrees and − 3 3 5 degrees are equivalent rotations. Hence, choice A is correct.

Adjacent, we will look at an case of identifying the image of a shape subsequently a rotation nigh the origin.

Example three: Identifying the Image of a Shape subsequently a Rotation about the Origin

If triangle 𝐴 𝐵 𝐶 is rotated past 𝑅 ( 𝑂 , ix 0 ) ∘ , which triangle would correspond its final position?

Answer

𝑅 ( 𝑂 , 9 0 ) ∘ indicates a rotation of 90 degrees counterclockwise virtually the origin. Visually, it is possible to see that a shape in the summit-right part of the graph (i.e., the first quadrant) would move to the top-left part (i.e., the second quadrant) afterward a xc-caste counterclockwise rotation, then the correct reply is either option B or E, which have figures in the second quadrant. From there, nosotros can run across that choice B is a reflection in the 𝑦 - a x i s rather than a rotation. Option E is correctly rotated 90 degrees counterclockwise from the original figure.

We can besides prove this mathematically, every bit this rotation tin likewise be expressed through the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) .

Take betoken 𝐴 from the preimage. Information technology is located at ( ane , 1 ) . After the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) , point 𝐴 ′ should be located at ( − i , i ) . Thus, we can narrow our choices down to option B or pick E, as those are the merely choices with point 𝐴 ′ at ( − 1 , 1 ) .

Next, consider point 𝐵 from the preimage. It is located at ( 3 , i ) . Afterwards the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) , point 𝐵 ′ should be located at ( − ane , three ) . Between options B and Eastward, only option E has point 𝐵 ′ at ( − i , 3 ) .

We may besides consider point 𝐶 from the preimage, which is located at ( ane , ii ) . Afterward the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) , point 𝐶 ′ should be located at ( − two , 1 ) . Option E does indeed show 𝐶 ′ at this location.

Hence, the right answer is pick Eastward, every bit this is the only choice where points 𝐴 ′ , 𝐵 ′ , and 𝐶 ′ all match the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) , representing a ix 0 ∘ counterclockwise rotation near the origin.

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

Example 4: Rotating a Triangle near the Origin

Determine the coordinates of the vertices' images of triangle 𝐴 𝐵 𝐶 after a counterclockwise rotation of 2 seven 0 ∘ effectually the origin.

Answer

A counterclockwise rotation of 270 degrees about the origin, which tin can be notated as 𝑅 ( 𝑂 , 2 7 0 ) ∘ , can exist represented by the coordinate transformation ( 𝑥 , 𝑦 ) → ( 𝑦 , − 𝑥 ) . To find the image of the shape after the rotation, nosotros can use this transformation to each of its vertices in turn.

Applying this coordinate transformation to point 𝐴 , the coordinates of which are ( − vii , − 3 ) , gives us 𝐴 ′ at ( − 3 , seven ) .

Applying this coordinate transformation to bespeak 𝐵 , the coordinates of which are ( − three , − 4 ) , gives us 𝐵 ′ at ( − 4 , 3 ) .

Applying this coordinate transformation to point 𝐶 , the coordinates of which are ( − 6 , − 5 ) , gives u.s.a. 𝐶 ′ at ( − v , half-dozen ) .

To check our work, we can visualize where these points would fall on the graph, and nosotros tin see that our new points lie in the 2d quadrant. This matches our prior knowledge that a 270-caste rotation well-nigh the origin would motion a shape from the third quadrant to the second quadrant.

Hence, the coordinates of the vertices' images of triangle 𝐴 𝐵 𝐶 later on a counterclockwise rotation of ii 7 0 ∘ around the origin are 𝐴 ′ ( − iii , 7 ) , 𝐵 ′ ( − four , 3 ) , and 𝐶 ′ ( − 5 , 6 ) .

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

Example v: Agreement the Properties of Rotation

In the figure, 𝐴 𝐵 has been rotated 9 0 ∘ counterclockwise about the origin. Is the length of the image 𝐴 ′ 𝐵 ′ resulting from this transformation greater than, less than, or the same as the length of 𝐴 𝐵 ?

Answer

Rotations are a "rigid transformation," which means that distances betwixt 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 ′ 𝐵 ′ is the aforementioned as the length of 𝐴 𝐵 .

Allow u.s. finish by recapping some key points from the explainer.

Central Points

  • A rotation of 𝑥 degrees is equivalent to a rotation of ( 𝑥 ± 3 half-dozen 0 ) degrees.
  • Coordinate transformations can be used to discover the images of rotated points as follows:
    • A rotation of xc degrees counterclockwise well-nigh the origin is equivalent to the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑦 , 𝑥 ) .
    • A rotation of 180 degrees counterclockwise about the origin is equivalent to the coordinate transformation ( 𝑥 , 𝑦 ) → ( − 𝑥 , − 𝑦 ) .
    • A rotation of 270 degrees counterclockwise about the origin is equivalent to the coordinate transformation ( 𝑥 , 𝑦 ) → ( 𝑦 , − 𝑥 ) .
    • A rotation of 360 degrees about the origin is equivalent to a rotation of 0 degrees and both are equivalent to the coordinate transformation ( 𝑥 , 𝑦 ) → ( 𝑥 , 𝑦 ) .
  • Line segments and shapes tin can be rotated by applying coordinate transformations to each of their endpoints or vertices.
  • To confirm if an paradigm on a coordinate plane is a rotation of a given preimage, we can utilize the advisable coordinate transformation to each point from the preimage and then verify if each point matches the corresponding betoken in the prototype.
  • Rotations are rigid transformations, which means that distances are preserved through the transformation.

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Source: https://www.nagwa.com/en/explainers/836184076070/

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