D AY 11 – I MAGES UNDER RIGID MOTION
I NTRODUCTION When a car moves from one place to another, its position changes but its size and shape remain unchanged. In a similar manner, when a plane figure undergoes a transformation such as a rotation, a reflection or a translation, both the pre- image and the image have the same size and shape. That is, there is no change in both size and shape. In this lesson we are going to look at the geometric description of such transformations to transform plane figures.
V OCABULARY 1. Isometry A transformation that leaves the lengths and angle measures of both the pre-image and the image unchanged. It is also called an isometric transformation . 2. 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. A rigid motion is simply an isometry.
3. Congruent figures These are figures that have the same size and shape. They can be mapped onto each other by one or more rigid motions. 4. Orientation of a plane figure The description of how the figure is placed in a plane and the arrangement of points on it after undergoing a transformation.
R IGID MOTIONS ON A PLANE A transformation is referred to as a rigid motion if both the pre-image and the image have the same size and shape. This means that angle measure and distance is preserved. It is also referred to as a rigid transformation, an isometry or a congruence transformation. There are three basic rigid motions: Reflections Rotations Translations Glide reflections
Properties of rigid motion 1. They preserve angle measure of figures. 2. They preserve distance in the figures.
Translations A translation is a rigid motion because the pre- image is slid onto the image and they both have the same size and shape.
Example 1 Triangle PQR is slid to the right as shown. P P R Q R Q
Explanation The pre-image and image are identical, they both have the same shape and size. The lengths and angle measures do not change. 𝑄𝑅 = 𝑄 𝑅 , 𝑅𝑆 = 𝑅 𝑆 , 𝑄 and 𝑆𝑄 = 𝑆 ∠𝑅 = ∠ 𝑅, ∠𝑄 = ∠ 𝑄, ∠𝑆 = ∠ 𝑆 This is a rigid motion.
Reflections When a figure is flipped, the image and the pre- image have the same size and shape. A reflection is also a rigid transformation.
Example The square ABCD has been flipped onto the other side of the line of reflection. A D 𝐸 𝐵 𝐷 B C 𝐶
The pre-image and the image have the same size and shape. The length of the sides and the angle measure have been preserved. This is a rigid motion. Note that the orientation of the square has changed after the flip.
Glide reflections A glide reflection is a transformation where the pre-image is reflected and translated parallel to the line of reflection. The order does not matter. Reflection Translation This is clearly a rigid motion.
Properties of a glide reflection: 1. It preserves distances 2. It preserves angles 3. It changes the figure’s orientation
Rotations When an image is turned about a fixed point, the image and the pre-image have the same size and shape. This implies that a rotation is a rigid motion.
The triangle has been turned about the fixed point O to the new position. It is clear, that both the pre- image and the image have the same size and shape. Pre-image Image O
Example Determine whether the given transformation is rigid or not. Solution The transformation is not rigid. The both the pre- image and the image have the same shapes but different sizes. The angle measure is preserved but the lengths are not preserved.
HOMEWORK Sketch the image of the trapezoid FGHI after a slide to the right labelling its vertices using prime symbol. State whether it is a rigid motion or not. F I H G
A NSWERS TO HOMEWORK F I H G It is a rigid motion.
THE END
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