1. An Isometry is a transformation that preserves length.
What is an Isometry?
An Isometry is a transformation that preserves length.

2. What transformations are Isometries?
Type
Diagram
A translation moves a figure left, right, up, or down
Preimage
Image
A reflection moves a figure across its line of reflection to create its mirror image.
A rotation moves a figure around a given point.

3. Translation/ how to Find point A and
Translate ABC according to the rule (x,y) (x+6,y)
Find point A and
Find point B and
Find point C and
6 Units A A’
count 6 units to the right. Plot point A’. B’ B C C’
count 6 units to the right. Plot point B’.
count 6 units to the right. Plot point C’.

4. Translation Rules To the right: Add to the X-coordinate
To the left: Subtract from the X-coordinate
Up: Add to the Y-coordinate
Down: Subtract from the Y-coordinate

5. 9 units to the right and 5 units down
Translation example: Translate point P (3, 2) according to the rule <X+9, Y-5>
9 units to the right and 5 units down
Add 3 to the x-coordinate = 12
Subtract 5 from the y-coordinate. 2 – 5 = -3
Therefore the coordinates of the image
P’ are (12, -3)

6. Reflection / how to Reflect ABC across the y-axis.
Count the number of units point A is from the line of reflection.
5 Units
5 Units A A’
Count the same number of units on the other side and plot point A’.
2 Units
2 Units B B’
3 Units
3 Units
Count the number of units point B is from the line of reflection. C C’
Count the same number of units on the other side and plot point B’.
Count the number of units point C is from the line of reflection.
Count the same number of units on the other side and plot point C’.

7. Reflection Rules Over the X-axis: Over the Y-axis:
Change sign for Y-coordinate
Over the Y-axis:
Change sign for X-coordinate

8. Reflection Rules Over line Y=X: Switch x and y coordinates
Over Line Y= -X: Change sign for y-coordinate AND change signs
Over line Y=X: Switch x and y coordinates

9. Reflection practice
The coordinates of ABC are: (-5, -4), (-2, 3), (3, 1) Reflect ABC across the y-axis and then reflect it across the x-axis.
To reflect it across the y-axis just change the sign for x. The new coordinates are: (5, -4), (2, 3), (-3, 1)
To reflect it across the x-axis keep the x the same and change sign for y. The new coordinates are:
(-5, 4), (2, -3), (-3, -1)

10. Switch the x and y coordinates:
Reflection practice
The coordinates of ABC are: (-5, -4), (-2, 3), (3, 1) Reflect ABC across the line Y= -X
Switch the x and y coordinates:
(-4, -5), (3, -2), (1, 3)
2. Change the signs
(+4, +5), (-3, +2), (-1, -3)

11. STOP HERE We will cover rotations on Monday during class.

12. Rotate  ABO 180° about the origin.
Rotation Rules
To rotate a point 180°, just change the signs for both coordinates.
Rotate  ABO 180° about the origin.
A (2, 4) A’ (-2, -4)
B (5, 1) B’ (-5, -1)
O (0, 0) O’ (0, 0)

13. Rotation Rules
To rotate a point 90° or 270° switch the x and y coordinates AND add look up signs on coordinate plane:
(2, 5)
Rotate point P (5, -2) 90° counter clock wise
1) Switch x and y: (-2, 5)
(5, -2)
2) The image should be in quadrant I where both signs are POSITIVE therefore P’ = (2, 5)

14. Rotation Rules
To rotate a point 90° or 270° switch the x and y coordinates AND add look up signs on coordinate plane:
Rotate point M (-1, -3) 270° clock wise
1) Switch x and y: (-3, -1)
(3, -1)
(-1 , -3)
2) The image should be in quadrant IV where X is positive and y is negative then M’ = (3, -1)

15. Rotations practice
Point P (5, 8). Rotate 270° clockwise about the origin.
Point Z (-3, -6). Rotate 180° about the origin.
Rotate LMN, with coordinates L(3, 6), M(5, 9), N (7, 12), 90° clockwise about the origin.
P’ (-8, 5)
Z’ (-3, -6)
L’M’N’ (6, -3), (9, -5), (7, -12)

16. Practice A 270° clockwise rotation about the origin is the same as…
Rotate point P (-3, 7) 90° counter clock wise about the origin
A 90° counter clock wise rotation
A 270° counter clock wise rotation
P’ = ( -7, -3)