A translation, or glide, and a reflection can be performed one after the other to produce a transformation known as a glide reflection. A glide reflection is a transformation in which every point P is mapped onto a point P” by the following steps :

1. A translation maps P onto P’.

2. A reflection in a line m parallel to the direction of the translation maps P’ onto P”.

As long as the line of reflection is parallel to the direction of the translation, it does not matter whether we glide first and then reflect, or reflect first and then glide.

Example 1 :

Use the information below to sketch the image of ΔABC after a glide reflection.

P(- 1, – 3), Q(- 4, – 1), R(- 6, – 4)

Translation : (x, y) —-> (x + 10, y)

Reflection : in the x-axis.

Solution :

Begin by graphing ΔPQR. Then, shift the triangle 10 units to the right to produce ΔP’Q’R’.

Finally, reflect the triangle ΔP’Q’R’ in x-axis to produce ΔP”Q”R”.

In the above example, try reversing the order of the transformations. Notice that the resulting image will have the same coordinates as ΔP”Q”R” above. This is true because the line of reflection is parallel to the direction of the translation.

When two or more transformations are combined to produce a single transformation, the result is called a composition of the transformations.

The composition of two (or more) isometries is an isometry.

Note :

Because a glide reflection is a composition of a translation and a reflection, this theorem implies that glide reflections are isometries. In a glide reflection, the order in which the transformations are performed does not affect the final image. For other compositions of transformations, the order may affect the final image.

Example 2 :

Sketch the image of AB after a composition of the given rotation and reflection.

A(2, – 2) and B(3, – 4)

Rotation : 90° counterclockwise about the origin

Reflection : in the y-axis

Solution :

Begin by graphing AB. Then, rotate the segment 90° counterclockwise about the origin to produce A’B’.

Finally, reflect the segment A’B’ in y-axis to produce A”B”.

Example 3 :

Repeat the above example given on finding the image of a composition, but switch the order of the composition by performing the reflection first and the rotation second. What do you notice?

Solution :

Begin by graphing AB. Then, reflect the segment in y-axis to produce A’B’.

Finally, rotate the segment A’B’ 90° counterclockwise about the origin to produce A”B”.

Instead of being in Quadrant II, as given in the example on finding the image of a composition, the image is in Quadrant IV.

The order which the transformations are performed affects the final image.

Example 4 :

Describe the composition of transformations in the diagram.

Solution :

Two transformations are shown. First figure PQRS is reflected in the line x = 2 to produce figure P’Q’R’S’. Then figure P’Q’R’S’ is rotated 90° clockwise about the point (2, 0) to produce P”Q”R”S”.

Example 5 :

The mathematical game pentominoes is a tiling game that uses twelve different types of tiles, each composed of five squares. The tiles are referred to by the letters they resemble. The object of the game is to pick up and arrange the tiles to create a given shape. Use compositions of transformations to describe how the tiles below will complete the 6 X 5 rectangle.

Solution :

Step 1 :

To complete part of the rectangle, rotate the A tile 90° clockwise, reflect the tile over a horizontal line, and translate it into place.

Step 2 :

To complete the rest of the rectangle, rotate the D tile 90° clockwise, reflect the tile over a vertical line, and translate it into place.

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