In this explainer, we will learn how to rotate points, line segments, and shapes about given points.
In geometry, we often use the congruence of shapes, angles, and line segments to prove results. This means that we want to find methods of checking the congruence of these objects. Often, this involves comparing the properties of two of these objects, but this is not the only way of studying congruence.
We can also ask, βWhat transformation can we apply to an object that does not affect its size or shape?β If a transformation does not affect these properties, then we can apply the transformation to any object to construct a congruent object. In particular, if we can show that two objects differ by one or more of these transformations, then we can conclude they are congruent.
There are many different transformations that do not affect the size or shape of an object; we can slide the object around, changing its position (translation), we can find a mirror image of the object (reflection), and we can also rotate the object about a point (rotation).
We will focus on rotating objects about a point. To start this, letβs consider exactly what rotation is. A rotation is defined by the center of the rotation and the amount of the rotation. In particular, we rotate each point around a circle centered at the center of the rotation. For example, letβs say we have the following rotation of a point centered at .
We call the image of a point after rotation . We can also add more prime symbols to represent further rotations. We rotate point along the circle of radius centered at .
This means that and are radii of the same circle, so they have the same length. This is true for any point after any rotation.
We can describe the direction of a rotation in two ways; clockwise and counterclockwise. Clockwise means the rotation is in the same direction that the hands of a clock move in; counterclockwise is the opposite direction.
We can also note that the rotation is given in a counterclockwise direction.
To describe the size of the rotation, we need a measure of how far a point moves after its rotation. We might be tempted to do this by measuring the length of the arc between the point and its image or the distance between the point and its image. However, neither of these measures will work since different points move different lengths.
Instead, we can measure the amount of rotation by considering the measure of .
This will tell us the size of the rotation to rotate onto . We can also use the size of the rotation to tell us the direction. We can say that a counterclockwise rotation is a positive rotation and that a clockwise rotation is a negative rotation.
For example, if we find that , then we can say that this is a counterclockwise rotation centered at . We write this . We can represent the clockwise rotation of the same measure as .
We can define all of this formally as follows.
Definition: Rotation about a Point
To rotate a point about a point , we move along the circle of radius centered at . We call the image of after this rotation .
We can rotate about a specific distance and direction by specifying the measure of , where we use a positive value to represent a counterclockwise rotation and a negative value to represent a clockwise rotation.
We use the notation to represent a rotation centered at of measure . The center of a rotation is the only point always fixed under rotation.
Letβs now see an example of using this notation to determine the image of a point after a given rotation.
Example 1: Rotating a Point about Another Point
Which point represents point after a rotation of ?
- Point
- Point
- Point
- Point
- Point
Answer
We start by recalling that the notation tells us that the rotation is centered at and that the measure of the rotation is . In fact, since this value is positive, we know that this rotation is counterclockwise.
A rotation centered at that is counterclockwise will move along the circle centered at of radius to a point such that .
We see that and that this is in a counterclockwise direction.
Hence, the answer is point .
Before we move on to our next example, we can consider some of the properties of rotations. For example, we stated at the start of the explainer that rotations do not affect the size or shape of the rotated object. We should justify this to make sure that it is true.
To do this, letβs first consider what happens when we rotate a line segment Β about . We can note that we can rotate a line segment by just rotating its endpoints, so the rotation of will be . This allows us to compare the lengths of these line segments.
We want to show that , and we can do this by proving the congruence of triangles and . We note that and since they are radii of the same circles. We can show that by noting that both sides of angle are rotated the same amount. Therefore, by the SAS criterion, and we can conclude that .
Hence, rotations do not affect the lengths of line segments or the measures of angles. In fact, since we can rotate a line segment by just rotating its endpoints, we can rotate any polygon by rotating its vertices and have a congruent polygon.
We can also rotate any point using a compass and protractor. For example, letβs say we want to rotate Β about a point . We can draw ray and then measure an angle clockwise from this ray. We can call this angle .
If we then trace a circle of radius centered at , we can note that the point of intersection between and the circle will be the image of after the rotation.
In our next example, we will determine the correct image of a triangle after a given rotation using a compass and protractor.
Example 2: Rotating a Triangle about a Point
When the green triangle is rotated about ,
which triangle is its new position?
Answer
We rotate a polygon about a point by first rotating all of its vertices. We recall that we rotate a point about by moving it along a circle centered at such that for any point in the shape , we have that the circle is of radius and that the measure of is equal to the measure of the rotation.
In this case, we want to rotate a triangle , and since this value is positive, we note that we will rotate the shape counterclockwise.
We want to determine the images of the vertices of the triangle, so letβs start with . Since is the center of rotation, it will not move under this rotation.
We can rotate vertices on triangle by considering circles centered at . We can first draw a ray and then measure another ray that is in the counterclockwise direction of , we call this . If we then sketch a circle centered at of radius , we note that is the point of intersection between the circle and ray and that .
Thus, the image of after the rotation is .
We can follow the same process for . We sketch ray and another ray from that is counterclockwise from . We then sketch a circle of radius centered at , and find that the point of intersection between the ray and the circle is point .
So, the image of is , the image of is , and the image of is .
Hence, the image of is , which is answer A.
In our next example, we will rotate a given triangle that makes up a tenth of a decagon.
Example 3: Understanding the Effect of a Rotation of a Triangle given the Angle of Rotation
What is the image of after a rotation?
Answer
We are asked to rotate Β about . To do this, we note that the given polygon is a regular decagon, so the angles around are all congruent. We can also note that the sum of the angles around must be . Thus, each angle at the center has a measure of . We then note that . So, we can find angles of measure on the diagram by rotating through 3 triangles in a clockwise direction.
We could now rotate the triangle by rotating its vertices using a compass and protractor; however, this is not necessary. We can note that each rotation of about will rotate the triangles clockwise such that they are coincident. One way of seeing this is to consider rotating a single line segment. For example, we can rotate Β about .
We then see that the image of this line segment is .
We can do the same for . If we rotate it about , we get .
Therefore, when we rotate Β about , the triangle will be rotated onto the third triangle clockwise from .
We see that this is .
In our next example, we will use the given image of a triangle after a rotation to determine the measure of the rotation.
Example 4: Finding the Measure of the Angle of Rotation
is the image of by a counterclockwise rotation of about . Find .
Answer
We begin by recalling that a rotation will rotate a line segment onto a congruent line segment whose endpoints are the images of the endpoints of the line segment under a rotation. In particular, we can note that this rotation is about , so will remain where it is after the rotation. Thus, is rotated onto .
We can determine the measure of the rotation by finding the measure of the angle between and . Since and the rotation is counterclockwise, we can see that the rotation has a measure of .
In our final example, we will consider rotations of a regular octagon.
Example 5: Comparing Two Rotations of the Same Line Segment
Consider the octagon shape with as its center.
- What is the final position of when it is rotated ?
- None of the answers are correct.
- What is the final position of when it is rotated ?
- None of the answers are correct.
- Is there a difference between the two rotations?
- Yes
- No
Answer
Part 1
To rotate a line segment, we recall that we can just rotate the endpoints. In other words, the image of will be . Letβs start by finding the image of . Since the measure of the rotation is positive, the rotation is counterclockwise. So, we want to rotate Β counterclockwise about .
We rotate a point Β about by finding point , which is counterclockwise on the circle centered at of radius such that . We can find this point on the diagram by noting that the distance between and any vertex of the regular octagon is the same. So, all of the vertices lie on a circle of radius centered at .
We can measure the angles at the center of the octagon by joining the center to all of the vertices and noting that the angles in the center are congruent and that their measures sum to . So, they are each of measure .
Thus, two of these angles combine to make , and so .
Thus, . We can follow the same reasoning to see that .
Hence, the image of after a rotation of is , which is answer C.
Part 2
We can also rotate Β using the same method. We need to note that since this value is negative, we rotate clockwise.
We see that the image of after a rotation of is and that the image of after a rotation of is .
Hence, the image of After a rotation of is , which is answer C.
Part 3
To answer this question, we need to consider what happens to a point after each rotation. Letβs say we rotate a point by about . does not necessarily need to be on the octagon; it could be any point in space, but we rotate it in the same way. We trace a circle centered at and find the point on the circle such that .
The reflex angle at will have a measure of since it makes a full turn with the other angle. If we measure this clockwise, then its measure is .
We can see that is also the image of after a rotation of .
Hence, the answer is that there is no difference between the transformations, which is answer B.
In the previous example, we say that a counterclockwise rotation of is the same transformation as a clockwise rotation of . In general, if the absolute values of the two angles of rotation of opposite signs add to give , then these rotations will be equivalent. In particular, this means that a counterclockwise rotation of is equivalent to a clockwise rotation of .
Letβs finish by recapping some of the important points from this explainer.
Key Points
- To rotate a point about a point , we move along the circle of radius centered at . We call the image of after this rotation .
- We can rotate about a specific distance and direction by specifying the measure of , where we use a positive value to represent a counterclockwise rotation and a negative value to represent a clockwise rotation.
- We use the notation to represent a rotation centered at of measure .
- If and are the images of and after a rotation, then is the image of after the same rotation and .
- We can rotate a polygon by rotating its vertices; the image of a polygon under a rotation is a congruent polygon.
- A counterclockwise rotation of is equivalent to a clockwise rotation of . In general, a counterclockwise rotation of is equivalent to a clockwise rotation of .
