In these slides you will learn the concepts and the basics of Translation, Reflection, Dilation, and Rotation. http://www.winpossible.com/lessons/Geometry_Translation,_Reflection,_Dilation,_and_Rotation.html

1. Translation, Reflection, Dilation, and Rotation
course offered by
www.winpossible.com

2. Translation, Reflection, Dilation, and Rotation
Translation
A transformation in which a geometric figure is moved to another location without
any change in size or orientation. Every translated figure, called the image of the
original figure, is congruent to original figure.
Example:
before translation
after translation

3. Translation, Reflection, Dilation, and Rotation (continued)
In the translation means ∆ABC to ∆A’B’C’,
A’
B’
Point A has moved to A’
Point B has moved to B’
Point C has moved to C’
A C’
∴ AA’ = BB’ = CC’ and AA’ || BB’ || CC’ B
∴∆ABC ≅ ∆A’B’C’
C

4. Translation, Reflection, Dilation, and Rotation (continued)
A translation has a horizontal component and a vertical component. If movement
with respect to the x-axis is l units and movement with respected to the y-axis is m
units, then for any point A(x, y), its coordinates become A’(x + l, y + m).
Example: Translate ∆ABC, 6 units left and 6 units down.
y
5
A (3, 4)
4
3 B
(5, 3)
2
1 C (1, 1)
-5 -4 -3 -2 -1 1 2 3 4 5 x
-1
A’ (-3, -2)
-2
-3
B’ (-1, -3)
(-5, -5) -4
C’

5. Translation, Reflection, Dilation, and Rotation (continued)
Reflection
A transformation in which a geometric figure is reflected across a line, creating a
mirror image. The line is called the line of reflection.
Example:
Axis of reflection

6. Translation, Reflection, Dilation, and Rotation (continued)
B B’
A A’
C C’
k
A is fixed point on line of reflection.
B’ is reflection of B in line k(B B’)
C’ is reflection of C in line k(C C’)

7. Translation, Reflection, Dilation, and Rotation (continued)
Reflection in a vertical line:
Reflection in a horizontal line:
Reflection in a diagonal line:

8. Translation, Reflection, Dilation, and Rotation (continued)
Reflection in the x axis
The x coordinates are the same and the y coordinates are the opposite.
y
5
4
A (1, 3)
3
C(3, 2)
2
1
B (1, 1)
-5 -4 -3 -2 -1 1 2 3 4 5 x
-1
-2
-3
-4

9. Translation, Reflection, Dilation, and Rotation (continued)
Reflection in the y axis
The y coordinates are the same and the x coordinates are the opposite.
y
4
A’ (-1, 3) A (1, 3)
3
C(5, 2)
2
C’ (-5, -2)
1
B’ (-1, 1) B (1, 1)
-5 -4 -3 -2 -1 1 2 3 4 5 x
-1
-2
-3
-4

10. Translation, Reflection, Dilation, and Rotation (continued)
Dilation
A transformation in which a figure is enlarged or reduced with respect to a point
called the center of dilation.
Dilation of a Geometric Figure
A transformation in which all dimensions are lengthened or shortened by a common
scale factor.
C
2
4
The smaller figure above is dilated with a scale factor of 2 and a center of dilation C
to produce the larger figure.

11. Translation, Reflection, Dilation, and Rotation (continued)
Rotation
A transformation in which a figure is rotated around a given point called the center
of rotation by a specified degree measure in a specified direction.
Example:
before rotation
angle of rotation = 90° clockwise
after rotation
center

12. Translation, Reflection, Dilation, and Rotation (continued)
Example:
Translate the triangle 4 units left and 5 units down.
5 y
C(5, 5)
A (2, 4)
4
3
2
1 B (3, 1)
-5 -4 -3 -2 -1 1 2 3 4 5 x
-1
-2
-3
-4

13. Translation, Reflection, Dilation, and Rotation (continued)
Example:
If the image of point C(0, -12) under a translation is C'(-5, -9), find the coordinates
of the image of point E(7, -8) under the same translation.

14. Translation, Reflection, Dilation, and Rotation (continued)
Example:
Find the images of points A(2, 3), B(-5, 2) and C(-1, 4) after a reflection in the
x-axis.
y
(-1, 4) C 4
3 A(2, 3)
B(-5, 2) 2
1
-5 -4 -3 -2 -1 1 2 3 4 5 x
-1
-2
-3
-4

15. Translation, Reflection, Dilation, and Rotation (continued)
Example:
Triangle ABC has coordinates A(-2, 0), B(6, 0), and C(4, 0). Find the coordinates of
the images of the vertices of the triangle after a reflection in the y-axis.

16. Translation, Reflection, Dilation, and Rotation (continued)
Example:
After a reflection in the x-axis, (10, -3) is the image of point E. What is the
original location of point E?

17. Translation, Reflection, Dilation, and Rotation (continued)
Example:
After a dilation, (45, 0) are the coordinates of the image of a point with
coordinates (5, 0). What are the coordinates of the image of (10, 25) after the
same dilation?

18. Translation, Reflection, Dilation, and Rotation (continued)
Example:
Translate the triangle 1 unit right and 3 units up, then rotate the image 180°
clockwise about the origin.
y
8
6
4
2
-10 -8 -6 -4 -2 2 4 6 8 10 x
-2
-4
-6
-8

19. Translation, Reflection, Dilation, and Rotation (continued)
Example:
Rotate the triangle in the figure 90° clockwise about the origin.
y
8
6
4
2
-10 -8 -6 -4 -2 2 4 6 8 10 x
-2
-4
-6
-8

20. Winpossible is a rapidly growing e-learning company based in California and New
York. Winpossible's math courses employ its patented ChalkTalk method or unique
ChalkTalkTM technology for presenting clear explanations on solving hundreds of
problems of varying levels of difficulty. Our methodology helps accurately diagnose
students’ performance levels, and allows them to focus content that is most relevant
to them. For more info, lots of free content, access to free math-homework help
and lots more on high-school math, please visit: www.winpossible.com