1. Invariance Under Transformations
Lesson 12.4
Invariance Under Transformations pp

2. Objectives: 1. To identify and prove properties of transformations.
2. To define isometry.

3. Invariance means “not varied” or “constant.”

4. Some transformations have invariant qualities
Some transformations have invariant qualities. If the preimage and image of a given transformation always share a certain characteristic, the transformation preserves that characteristic.

5. Reflections, rotations, and translations all preserve distance because the image is always exactly the same as the preimage.

6. Definition
An isometry is a transformation that preserves distance.

7. Identity transformation Composition of isometries
Reflection
Translation
Rotation
Identity transformation
Composition of isometries

8. Properties of Isometries
1. Distance is preserved.
2. Collinearity of points is preserved.
3. Betweenness of points is preserved.
4. Angle measure is preserved.
5. Parallelism is preserved.
6. Triangle congruence is preserved.

9. Theorem 12.1
Isometry Theorem. Every isometry can be expressed as a composition of at most three reflections.

10. Dilations do not preserve distance and are not isometries.

11. A transformation that preserves shape is a similarity while a transformation that preserves both size and shape is an isometry.

12. Properties of Dilations
A dilation preserves:
1. Collinearity of points.
2. Betweenness of points.
3. Angle measure.
4. Parallel lines.

13. Similarities and isometries are not the only types of transformations
Similarities and isometries are not the only types of transformations. It is possible to have a transformation with none of these invariant qualities.

14. For the next two slides, which of the six properties of isometries are preserved for T? Is T an isometry?

15. T translates all points of one half-plane two units right and all points of the other half-plane two units to the left.

16. T doubles the distance of each point from the line.

17. Homework pp

18. ►A. Exercises
Name the type of transformation illustrated by the pair of figures. Is it an isometry? Choose from (1) reflection, (2) translation, (3) rotation, (4) dilation, or (5) identity transformation. 1. P′ Q′ R′ P Q R

19. ►A. Exercises
Name the type of transformation illustrated by the pair of figures. Is it an isometry? Choose from (1) reflection, (2) translation, (3) rotation, (4) dilation, or (5) identity transformation. 3. C′ A′ B′ C B A

20. ►A. Exercises
Name the type of transformation illustrated by the pair of figures. Is it an isometry? Choose from (1) reflection, (2) translation, (3) rotation, (4) dilation, or (5) identity transformation. 5. N M N″ M″ N′ M′ l

21. ►A. Exercises
Name the type of transformation illustrated by the pair of figures. Choose from (1) reflection, (2) translation, (3) rotation, or (4) dilation. 7. A

22. 9. Reflect ABC in m; then reflect it in l.
►A. Exercises
Trace the following onto your paper and then find the image and identify the type of isometry.
9. Reflect ABC in m; then reflect it in l. A″ C″ B″ A C B l A B m C

23. 11. A subset of a line by an isometry
►B. Exercises
If each figure below is transformed as indicated, what must the image be and why?
11. A subset of a line by an isometry
A subset of a line, because collinearity is preserved.

24. 13. Parallel lines by a reflection
►B. Exercises
If each figure below is transformed as indicated, what must the image be and why?
13. Parallel lines by a reflection
Parallel lines, since parallelism is preserved by isometries.

25. 15. Point A rotated 90° around C then rotated another 90° around C
►B. Exercises
Draw the following transformation and, if possible, give the following composite transformations in simpler form.
15. Point A rotated 90° around C then rotated another 90° around C

26. ►B. Exercises
15. B′ A′ D′ A B C D B″ D″ A″
180° rotation around C

27. 16. ∆ABC reflected in l and then rotated 90° clockwise around B
►B. Exercises
Draw the following transformation and, if possible, give the following composite transformations in simpler form.
16. ∆ABC reflected in l and then rotated 90° clockwise around B

28. ►B. Exercises
16. A B C l C′ A′ B′ A″ B″ C″

29. ►B. Exercises
Draw the following transformation and, if possible, give the following composite transformations in simpler form.
17. AB enlarged from point P by a scale factor of 6 and then contracted from point P by a scale factor of ½

30. ►B. Exercises 17. enlarge from point P by a scale factor of 3 B′ B″ B

31. ■ Cumulative Review
Name a theorem that could be used to show that two segments are congruent if the segments are related to the figure as indicated. 21. quadrilateral

32. ■ Cumulative Review
Name a theorem that could be used to show that two segments are congruent if the segments are related to the figure as indicated. 22. triangle

33. ■ Cumulative Review
Name a theorem that could be used to show that two segments are congruent if the segments are related to the figure as indicated. 23. circle

34. ■ Cumulative Review
Name a theorem that could be used to show that two segments are congruent if the segments are related to the figure as indicated. 24. perpendiculars

35. ■ Cumulative Review
Name a theorem that could be used to show that two segments are congruent if the segments are related to the figure as indicated. 25. lines

36. ■ Cumulative Review
Name a theorem that could be used to show that two segments are congruent if the segments are related to the figure as indicated. 26. space