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Here are a few recommended readings before getting started with this lesson.
As stated at the beginning of this lesson, in Geometry, there are functions whose inputs and outputs are points. Such functions are called transformations.
A transformation is a function that changes a figure in a particular way — it can change the position, size, or orientation of a figure. The original figure is called the preimage and the figure produced is called the image of the transformation. A prime symbol is often added to the label of a transformed point to denote that it is an image.
Transformations are sometimes expressed as a mapping because they map the inputs to the outputs. Note that an input can be a single point.Both P and P′ have the same side lengths and angle measures. For this reason, it can be established that T does not affect the shape of P, which means that T preserves the side lengths and angle measures.
Use a ruler to find the side lengths of both polygons. To find the angle measures, use a protractor. Make a table comparing the dimensions of P and P′.
With the aid of a ruler, the side lengths of both polygons can be found.
Additionally, with the help of a protractor, the angle measures of both polygons can be determined.
It is beneficial to summarize the information about the side lengths and angle measures of both polygons in a table.
Dimensions of P | Dimensions of P′ |
---|---|
AB=2.1 cm | A′B′=2.1 cm |
BC=2.4 cm | B′C′=2.4 cm |
CD=2 cm | C′D′=2 cm |
AD=1.5 cm | A′D′=1.5 cm |
m∠A=74∘ | m∠A′=74∘ |
m∠B=92∘ | m∠B′=92∘ |
m∠C=60∘ | m∠C′=60∘ |
m∠D=134∘ | m∠D′=134∘ |
As it can be seen in the table, both polygons P and P′ have the same side lengths and angle measures. Therefore, it can be concluded that T does not affect the shape of P. In fact, T only affects the position of the polygon.
The transformation T preserves side lengths and angle measures.
It is important to note that the conclusion does not depend on the polygon but on the effect of T on the polygon. By transforming different polygons, the same conclusion can be obtained.
For each triangle, use a ruler and a protractor to find the side lengths and angle measures, respectively. Make a table comparing the dimensions of △JKL and △J′K′L′.
With the aid of a ruler, the side lengths of both triangles can be found.
Next, with the help of a protractor, the angle measures of both triangles can be found.
The dimensions found can be summarized in a table.
Dimensions of △JKL | Dimensions of △J′K′L′ |
---|---|
JK=2.2 cm | J′K′=2.2 cm |
KL=3.5 cm | K′L′=3.5 cm |
JL=1.7 cm | J′L′=1.7 cm |
m∠J=127∘ | m∠J′=127∘ |
m∠K=23∘ | m∠K′=23∘ |
m∠L=30∘ | m∠L′=30∘ |
As the table shows, both △JKL and △J′K′L′ have the same side lengths and angle measures. Therefore, T does not affect the shape of the polygon. Consequently, Magdalena is correct.
The transformation T preserves side lengths and angle measures.
It is important to note that the conclusion does not depend on the polygon chosen. Instead, the conclusion depends on the effect of T on the polygon. By using different polygons, the same conclusion can be obtained.
Notice that the two transformations previously studied share the same property. The transformations neither affected the size nor the shape of the polygon. Still, they did affect the polygon's position on the plane. These types of transformations are called rigid motions.
Because rigid motions preserve distances, there are two properties that can be inferred from the definition.
A rigid motion preserves the side lengths and angle measures of a polygon. As a result, a rigid motion maintains the exact size and shape of a figure. Still, a rigid motion can affect the position and orientation of the figure.
Ali is correct, T is a rigid motion. Magdalena could be confused because the transformation changes the orientation of the quadrilateral.
Using a ruler, the side lengths of both quadrilaterals can be found.
Next, with a protractor, the angle measures can be found.
In the table, all the dimensions found are summarized putting on the same row the dimensions of corresponding parts. For example, AB and A′B′ will be in the same row.
Dimensions of ABCD | Dimensions of A′B′C′D′ |
---|---|
AB=2.8 cm | A′B′=2.8 cm |
BC=2.2 cm | B′C′=2.2 cm |
CD=1.8 cm | C′D′=1.8 cm |
AD=1.9 cm | A′D′=1.9 cm |
m∠A=72∘ | m∠A′=72∘ |
m∠B=80∘ | m∠B′=80∘ |
m∠C=88∘ | m∠C′=88∘ |
m∠D=121∘ | m∠D′=121∘ |
As the table shows, both quadrilaterals ABCD and A′B′C′D′ have the same side lengths and angle measures. This means that T does not affect the shape of the polygon. Consequently, Ali is correct.
The transformation T is a rigid motion.
Although T is a rigid motion, notice that the orientation of the preimage and the image are not the same. In the preimage, the vertices from A to D are positioned counterclockwise, while in the image, they are positioned clockwise.
This fact could be what made Magdalena think that T is not a rigid motion. Notice that the transformations studied before preserve orientations.
Mapping the preimage onto the image would necessitate folding the tracing paper along the line of reflection. However, by doing this the tracing paper peels off the sketch pad making a three-dimensional movement.
Consider a triangle with vertices A(-2,-1), B(2,1), and C(0,2).
Original Vertex | Add 3 to Each y-Coordinate | New Vertex |
---|---|---|
A(-2,-1) | (-2,-1+3) | A′(-2,2) |
B(2,1) | (2,1+3) | B′(2,4) |
C(0,2) | (0,2+3) | C′(0,5) |
Original Vertex | Double each x-Coordinate | New Vertex |
---|---|---|
A(-2,-1) | (2(-2),-1) | A′(-4,-1) |
B(2,1) | (2(2),1) | B′(4,1) |
C(0,2) | (2(0),2) | C′(0,2) |
Original Vertex | Multiply Each y-Coordinate by -1 | New Vertex |
---|---|---|
A(-2,-1) | (-2,-1(-1)) | A′(-2,1) |
B(2,1) | (2,1(-1)) | B′(2,-1) |
C(0,2) | (0,2(-1)) | C′(0,-2) |
Original Vertex | Taking the Absolute Value of Each y-Coordinate | New Vertex |
---|---|---|
A(-2,-1) | (-2,∣-1∣) | A′(-2,1) |
B(2,1) | (2,∣1∣) | B′(2,1) |
C(0,2) | (0,∣2∣) | C′(0,2) |
foldedthe part of the triangle that is below the x-axis along the x-axis. Consider also the points P(-1,-0.5) and Q(-1,0.5).
These two points have the same image under this operation — P′(-0.5,0.5) and Q′(-0.5,0.5). This means that this is not a one-to-one operation. Consequently, it is not even a transformation!
Consider the fact that two or more functions can be applied one after the other to an input. Similarly, two or more transformations can be applied one after the other to a preimage.
A composition of transformations, or sequence of transformations, is a combination of two or more transformations. In a composition, the image produced by the first transformation is the preimage of the second transformation. The notation is similar to the notation used for functions in algebra.
If the transformations are rigid motions, the composition is also a rigid motion.Magdalena and Ali were each given a rigid motion that they have to apply to pentagon ABCDE one after the other in a particular order.
Their teacher said that after applying both transformations in the correct order, the image of B(-3,-2) is B′′(4,-4). Who should apply the transformation first?
If the point (x,y) is rotated 90∘ counterclockwise about the origin, its new coordinates will be (-y,x).
Preimage | Translation Followed by Rotation | Rotation Followed by Translation |
---|---|---|
B(-3,-2) | B′′(3,-1) | B′′(4,-4) |
C(-1,-4) | C′′(5,1) | C′′(6,−2) |
The teacher said that performing the transformations in the correct order maps B(-3,-2) onto B′′(4,-4). This means that the correct order is the rotation followed by the translation. Therefore, Ali goes first. If the transformations are not performed in the correct order, the image of C(-1,-4) is C′′(5,1).
Keep in mind that transformations transcend beyond paper and can be used for many purposes. In the real world, transformations occur everywhere. For example, in nature, lakes perform reflections of landscapes.
Also, different types of transformations can be seen by observing a car, either in rest or in movement.