DAY 48 COMPOSITION OF RIGID MOTIONS INTRODUCTION In

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DAY 48 – COMPOSITION OF RIGID MOTIONS

DAY 48 – COMPOSITION OF RIGID MOTIONS

INTRODUCTION In the previous lessons, we took an in-depth study on the basic rigid

INTRODUCTION In the previous lessons, we took an in-depth study on the basic rigid motions, basically, rotations, reflections, translations and glide reflections, including their key properties. It is possible to perform a combination of two or more rigid motions on a given pre-image, that is, using the image of the first rigid motion as the preimage for the next rigid motion. In this lesson, we will show that a combination of two rigid motions is also a rigid motion and when a half-turn and a reflection are combined on a given pre-image, the effect is the same as transforming the pre-image through a glide reflection.

VOCABULARY 1. Rigid motion A transformation which changes the position of a plane figure

VOCABULARY 1. Rigid motion A transformation which changes the position of a plane figure without changing the figure’s shape or size. It is also called a rigid transformation. 2. Glide reflection A transformation where the pre-image is reflected and translated parallel to the line of reflection. The order does not matter.

3. Composition of rigid motions If a certain rigid motion is applied to a

3. Composition of rigid motions If a certain rigid motion is applied to a plane figure and then one or more rigid motions applied to its image, the result is called a composition of rigid motions. It is also referred to as a sequence of rigid motions

We know that under any type of rigid motion: 1. Lengths of line segments

We know that under any type of rigid motion: 1. Lengths of line segments on the plane figure are preserved 2. Angle measures of the plane figure are preserved 3. Collinear points remain collinear after the transformation 4. Parallel lines remain parallel after the transformation

In a nutshell, the image is exactly the same size and shape as the

In a nutshell, the image is exactly the same size and shape as the pre-image. Considering these key properties of rigid motion, we can show that any composition of two rigid motions is also a rigid motion. The rigid motions can be of one type or different types.

THE COMPOSITION OF TWO RIGID MOTIONS A rigid motion preserves lengths and angle measures,

THE COMPOSITION OF TWO RIGID MOTIONS A rigid motion preserves lengths and angle measures, implying that regardless of the number of rigid motions performed, the final image will have the same size and shape as the original pre-image. In a case of two rigid motions, lengths and angle measures will be preserved after the two rigid motions on the pre-image.

Theorem: The composition of two rigid motions is also a rigid motion. Explanation: This

Theorem: The composition of two rigid motions is also a rigid motion. Explanation: This implies that if we perform any rigid motion on a given pre-image, then we treat the resulting image as our new pre-image and then perform the second rigid transformation, the resulting image will be the same shape and size are the original pre-image. Shape and size is preserved after the two transformations. We can use any two consecutive rigid motions on a given plane figure to show this.

Let us perform a rotation, followed by a translation of the triangle below. Note

Let us perform a rotation, followed by a translation of the triangle below. Note that both a translation and a rotation are rigid motions. We will compare the final image to the original pre-image afterward.

 5 4 3 2 1 A 1 2 3 4 5 B C

5 4 3 2 1 A 1 2 3 4 5 B C 6 7

THE COMPOSITION OF A HALF-TURN AND REFLECTION

THE COMPOSITION OF A HALF-TURN AND REFLECTION

Theorem: The composition of a half-turn and a reflection is a glide reflection if

Theorem: The composition of a half-turn and a reflection is a glide reflection if the center of rotation is not on the line of reflection. Note In a case where the center is on the line of reflection, the composition becomes a glide reflection where the translation involved is of zero distance, that is a mere reflection. Thus, this can be termed as a special case of the line.

 y 5 4 3 2 1 A 0 1 2 3 4 5

y 5 4 3 2 1 A 0 1 2 3 4 5 B C 6 7 x

 5 4 y 3 2 1 A B C 0 1 2 3

5 4 y 3 2 1 A B C 0 1 2 3 4 5 6 7 x

 5 4 y 3 2 1 A 0 B C 1 2 3

5 4 y 3 2 1 A 0 B C 1 2 3 4 5 6 7 x

 K L M Mirror line, l

K L M Mirror line, l

HOMEWORK A D B C

HOMEWORK A D B C

ANSWERS TO HOMEWORK

ANSWERS TO HOMEWORK

THE END

THE END