1. Geometry
January 30, 2015

2. Not all objects have a line of symmetry
A line of symmetry can be drawn through an object to create two congruent halves that are mirror images of each other.
Not all objects have a line of symmetry
Objects can have more than one line of symmetry

3. Examples of objects with symmetry: 5 lines 4 lines 1 line

4. Dilations
A dilation is a shrink or stretch of an object. Doing this does NOT create an isometry but instead creates a similar shape to the original

5. Dilations
For coordinate dilations we multiply the vertices by a scalar
𝑥,𝑦 𝑤𝑖𝑡ℎ 𝑎 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑎𝑐𝑡𝑜𝑟 𝑐 𝑖𝑠 (𝑐𝑥,𝑐𝑦)
A dilation is a stretch when the dilation factor is greater than 1
A dilation is a shrink when the dilation factor is less than 1

6. Dilations with matrices
We apply dilations with matrices by using a scalar multiplication:
Example:
−6 3 2 −8 with a dilation of factor 4
4 −6 3 2 −8
− −32

7. Example 1
−33 −18 9 −9 with a dilation by factor −33 −18 9 − −11 −6 3 −3

8. HOMEWORK
Assignment 7-5

9. Composition of transformations
A composition of transformations is applying more than one transformation in the order that they are listed.
Does the order really matter?
Yes
Here is an example:
Take the following points 𝐴(3,6) and 𝐵(5,9)
Apply the following two transformations a reflection over the 𝑥−𝑎𝑥𝑖𝑠 and a translation (𝑥, 𝑦) −> (𝑥−2, 𝑦−1)

10. Compositions of Transformations
First lets try it by doing the reflection THEN the translation
𝐴 3,6 𝑎𝑛𝑑 𝐵 5,9 reflect over 𝑥−𝑎𝑥𝑖𝑠 to 𝐴′ 3, −6 𝑎𝑛𝑑 𝐵′ 5, −9
Then translate (𝑥, 𝑦) −> (𝑥−2, 𝑦−1) to 𝐴′′ 1,−7 𝑎𝑛𝑑 𝐵′′(3,−10)
Now lets try it by doing the translation first THEN doing the reflection
𝐴 3,6 𝑎𝑛𝑑 𝐵(5,9) translated 𝑥,𝑦 → 𝑥−2,𝑦−1 to A′ 1, 4 𝑎𝑛𝑑 𝐵′ 3, 8
Then reflect over the 𝑥−𝑎𝑥𝑖𝑠 to 𝐴′′ 1, −4 𝑎𝑛𝑑 𝐵′′(3, −8)
As you can see the two do NOT result in the same A ′′ 𝑎𝑛𝑑 𝐵′′ values which indicate that the order does matter.

11. HOMEWORK
Assignment 7-6