Rotation On A Coordinate Plane

In this explainer, nosotros will acquire how to observe the vertices of a shape after it undergoes a rotation of xc, 180, or 270 degrees about the origin clockwise and counterclockwise.

Let us start by rotating a point. Retrieve that a rotation by a positive caste value is divers to exist in the counterclockwise management.

Take the point , which is located in the summit-right part of the -plane (i.east., the outset quadrant). We will call this point .

Rotating point past 90 degrees virtually the origin gives us point at coordinates . This is fabricated clearer past connecting line segments from the origin to points and , from which we tin can run into that a right angle is formed.

Notice the reoccurrence of the 3 and 4 from the coordinates of point . In fact, all rotations nigh the origin in multiples of 90 degrees volition follow similar patterns. In general terms, rotating a signal with coordinates by ninety degrees nigh the origin will result in a point with coordinates .

Now, consider the point when rotated by other multiples of 90 degrees, such as 180, 270, and 360 degrees. We volition add points and to our diagram, which stand for point rotated past 180 and 270 degrees counterclockwise respectively. Find that rotating point past 360 degrees will bring information technology back to where it started, to the coordinates .

The rotation of a point by 180 degrees is represented by the coordinate transformation . The rotation of a point by 270 degrees is represented by the coordinate transformation . The rotation of a signal by 360 degrees does not change its coordinates, and such a rotation can be represented by the coordinate transformation .

We also annotation that all rotations nigh the same point that differ by a multiple of 360 degrees are equivalent. This is considering rotating by 360 degrees brings us back to where nosotros started. For case, rotating by is the same every bit rotating by and and then by ; since the rotation of does non change the coordinates, rotating by is the aforementioned as rotating by only. We must as well note that these rotations are only equivalent when they are about the same point. Rotations around dissimilar points can never be equivalent unless the rotation is degrees (or equivalent to a rotation of degrees).

Furthermore, a negative (clockwise) rotation can always be reexpressed as a positive (counterclockwise) rotation. As an example, we will detect a positive equivalent to a rotation of . Since rotations that differ by a multiple of are equivalent, we tin can add together to the rotation, which gives us in total. Hence, a rotation of is equivalent to a rotation of , and, therefore, both rotations can be expressed every bit the coordinate transformation .

Belongings: Rotations by Multiples of ninety Degrees nearly the Origin

For a indicate with coordinates , the following is true:

A rotation of 90 degrees results in a point with coordinates .
A rotation of 180 degrees results in a point with coordinates .
A rotation of 270 degrees results in a point with coordinates .
A rotation of 360 degrees results in a point with coordinates .
A rotation with a positive caste value indicates a counterclockwise rotation, and a rotation with a negative degree value indicates a clockwise rotation.
A rotation of is equivalent to a rotation of and therefore results in a point with coordinates .
A rotation of is equivalent to a rotation of and therefore results in a betoken with coordinates .
A rotation of is equivalent to a rotation of and therefore results in a bespeak with coordinates .
A rotation of is equivalent to a rotation of and therefore results in a point with coordinates .

Now that nosotros have introduced rotations of points, we will talk over 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 thing of applying the coordinate transformations to multiple points.

For example, consider triangle with vertices , , and . To rotate this triangle 180 degrees about the origin, we need to rotate each of its points according to the coordinate transformation . The rotated triangle volition have vertices , , and .

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

For case, if we are given the graph above and asked to verify that triangle is a rotation of triangle most the origin, we volition first take point . Applying the coordinate transformation gives us , confirming that these are the coordinates of point . After nosotros ostend this for points and and for points and , we can verify that triangle is indeed the image of triangle after a rotation about the origin.

Lastly, nosotros notation that all rotations share a special holding. 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 effigy is a transformation that preserves the distance betwixt each pair of points in the effigy.

A rigid transformation is sometimes referred to as an isometry.

For example, if the altitude from point to bespeak is equal to and they are rotated most the same point , so the distance between the new points and will still be . Rotations, reflections, and translations are examples of rigid transformations.

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

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

What is the image of under the transformation ?

Answer

Let us employ the coordinate transformation to each signal in . In each instance, we want to swap the - and -coordinates then reverse the sign of the get-go coordinate. Doing this gives us

Therefore, the image of under the transformation is given by the points , , , and .

The rotation appears to be a ninety-caste counterclockwise rotation. This matches our knowledge that such a rotation is represented by the coordinate transformation , which is the coordinate transformation practical in the problem.

Next, nosotros 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 rotation about the origin?

A rotation well-nigh the origin
A rotation nearly the origin
A rotation virtually the origin
A rotation almost the point
A rotation about the origin

Answer

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

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

While adding 360 degrees does not give us whatsoever of the options shown, if we subtract 360 degrees, we become

Therefore, 25 degrees and degrees are equivalent rotations. Hence, pick A is correct.

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

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

If triangle is rotated past , which triangle would represent its final position?

Answer

indicates a rotation of xc degrees counterclockwise about the origin. Visually, it is possible to see that a shape in the top-right function of the graph (i.e., the offset quadrant) would move to the top-left part (i.e., the 2d quadrant) later on a 90-degree counterclockwise rotation, and so the correct answer is either option B or E, which accept figures in the 2nd quadrant. From there, we can come across that option B is a reflection in the rather than a rotation. Choice Eastward is correctly rotated 90 degrees counterclockwise from the original figure.

We can also prove this mathematically, as this rotation can besides exist expressed through the coordinate transformation .

Accept bespeak from the preimage. It is located at . After the coordinate transformation , point should be located at . Thus, we can narrow our choices down to option B or option E, as those are the only choices with point at .

Next, consider signal from the preimage. It is located at . After the coordinate transformation , bespeak should exist located at . Between options B and E, only option Eastward has signal at .

We may likewise consider point from the preimage, which is located at . After the coordinate transformation , bespeak should be located at . Choice Eastward does indeed show at this location.

Hence, the correct answer is option E, as this is the only choice where points , , and all match the coordinate transformation , representing a counterclockwise rotation about the origin.

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

Instance iv: Rotating a Triangle most the Origin

Make up one's mind the coordinates of the vertices' images of triangle afterwards a counterclockwise rotation of around the origin.

Answer

A counterclockwise rotation of 270 degrees most the origin, which can be notated as , can be represented by the coordinate transformation . 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 , the coordinates of which are , gives us at .

Applying this coordinate transformation to signal , the coordinates of which are , gives u.s.a. at .

Applying this coordinate transformation to point , the coordinates of which are , gives us at .

To check our work, we tin visualize where these points would fall on the graph, and we can see that our new points lie in the 2d quadrant. This matches our prior cognition that a 270-degree rotation nigh the origin would move a shape from the 3rd quadrant to the second quadrant.

Hence, the coordinates of the vertices' images of triangle after a counterclockwise rotation of around the origin are , , and .

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

Example 5: Understanding the Properties of Rotation

In the figure, has been rotated counterclockwise about the origin. Is the length of the prototype resulting from this transformation greater than, less than, or the aforementioned every bit the length of ?

Answer

Rotations are a "rigid 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 2 endpoints and the distance between these 2 points is preserved through the transformation, the length of the prototype is the same every bit the length of .

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

Key Points

A rotation of degrees is equivalent to a rotation of degrees.
Coordinate transformations tin be used to find the images of rotated points as follows:
- A rotation of 90 degrees counterclockwise near the origin is equivalent to the coordinate transformation .
- A rotation of 180 degrees counterclockwise almost the origin is equivalent to the coordinate transformation .
- A rotation of 270 degrees counterclockwise about the origin is equivalent to the coordinate transformation .
- A rotation of 360 degrees most the origin is equivalent to a rotation of 0 degrees and both are equivalent to the coordinate transformation .
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 use the appropriate coordinate transformation to each point from the preimage and so verify if each point matches the corresponding point in the image.
Rotations are rigid transformations, which ways that distances are preserved through the transformation.