1. Day 13 – Effects of rigid motion

2. Introduction
So far we have discussed types of rigid motion and how to identify the respective rigid motions after examining the pre-image and the resulting image(s). In this lesson, we are going to learn how to predict the effect of a given rigid motion on a certain plane figure.

3. Vocabulary 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.
Orientation
The position of a plane figure and the arrangement of points in relation to one another after a transformation.
Collinear points
Points that lie on the same straight line.

4. Reflections, rotations, translations and glide reflections are all rigid motions. Rigid motions are also called rigid transformations because they preserve the shape and size of the pre-image after transformation.

5. Effects of rigid motion on a plane figure
Distance is preserved, the lengths of the line segments on the figure remain the same before and after transformation.
Angle measure is preserved, the sizes of the angles on the plane figure do not change after transformation.
Parallel lines on the figure remain parallel after transformation.
Collinearity is preserved, points remain on the same lines after the transformation.
Orientation of the figure may change or not depending on the type of transformation.

6. Translations:
They are rigid motions that slide a plane figure through a given distance in a fixed direction. A translation has the following effects on a plane figure:
1. All the points on the pre-image move the same distance in the same direction.
2. The image has exactly the same size and shape as the pre-image.
3. Angle measure remain the same after the translation.

7. 4. Parallel lines on the figure remain parallel after the translation
4. Parallel lines on the figure remain parallel after the translation. 5. Points remain on the same lines after the translation. 6. Orientation is preserved. 7. Parallel lines remain parallel.

8. Example 1 A translation T (c,d) : 𝑥,𝑦 → 𝑥+𝑐, 𝑦+𝑑 slides a figure c units in the x-coordinate followed by d units in the y-coordinate.

9. ∆𝑃𝑄𝑅 is translated to ∆ 𝑃 𝑄 𝑅P Q R Q

10. 1. The distances from the pre-image points to the image points are equal. P P= Q Q =R R 2. The line segments representing the distances from the pre-image points to the image points are parallel. P P ∥Q Q ∥R R 3. The corresponding line segment sides of the pre- image and the image are parallel 𝑃𝑄∥ 𝑃 𝑄 ; 𝑄𝑅∥ 𝑄 𝑅 ; 𝑃𝑅∥ 𝑃 𝑅

11. 4. The orientation remains the same after the translation
4. The orientation remains the same after the translation. The pre-image is lettered clockwise as PQR and the image too, as P Q R . 5. The angle measure is preserved: ∠𝑃=∠ 𝑃 , ∠𝑄=∠ 𝑄 and ∠𝑅=∠ 𝑅 6. Points remain on the same line segments.

12. Reflections A reflection over a given mirror line m, denoted as 𝑟 𝑚 is a transformation where: 1. Every point of the pre-image has an image that is the same distance from the line of reflection as the pre-image. 2. The image is on the opposite side of the line of reflection. 4. It is a rigid transformation because the image is the same shape and size as the pre-image. 5. Distance is preserved.

13. 6. Angle measure is preserved. 7. Parallel lines remain parallel. 8
6. Angle measure is preserved. 7. Parallel lines remain parallel. 8. Collinear points remain on the same lines. 9. Orientation changes. 10. The pre-image and the image are directly congruent.

14. Example 2 ∆𝑃𝑄𝑅 is reflected to ∆ 𝑃 𝑄 𝑅 over the mirror line 𝑚 Q R 𝑚

15. We note the following from the diagram: 1
We note the following from the diagram: 1. The distances from the pre-image points to the image points are different, they are not necessarily equal. P P≠ Q Q ≠R R 2. The line segments representing the distances from the pre-image points to the image points are are parallel. P P∥ Q Q ∥R R 3. The orientation has changed. The image is lettered clockwise PQR while the image is lettered

16. counterclockwise PQR. 4. The pre-image and the image are oppositely congruent. 5. A point lying on the line of reflection remains at the same position after reflection; the point becomes its own reflection.

17. Rotations A rotation of 𝜃 degrees denoted as 𝑅 (𝑐,𝜃) is a transformation that turns a plane figure about a fixed point, c, called the center of rotation. The center of rotation is usually the origin on the coordinate plane. Under a rotation: 1. The distances from the pre-image points to the image points are different, they are not necessarily equal. 2. The line segments representing the distances from the pre-image points to the image points are not parallel

18. 3. The orientation is preserved. 4. Angle measure is preserved. 5
3. The orientation is preserved. 4.Angle measure is preserved. 5. Parallel line segments remain parallel. 6. Points remain on the same lines.

19. Example 3 ∆𝑃𝑄𝑅 is rotated to ∆ 𝑃 𝑄 𝑅 through an angle 𝜃 clockwise about center O. Q P P 𝜃 Q R R O

20. We observe the following: 1. Distance is preserved 2
We observe the following: 1. Distance is preserved 2. Angle measure is preserved 3. Parallel line segments on the figure remain parallel 4. Collinear points remain parallel 5. Orientation is preserved 6. The line segments representing the distances from the pre-image points to the image points are not parallel

21. homework
Identify the type of rigid motion characterized by non-parallel line segments representing the distances from the pre-image points to the corresponding image points.

22. Answers to homework
Rotation

23. THE END