1. By Yena Shin, Nicola Licata, Mona Moshet & Andi Deng
Transformations
By Yena Shin, Nicola Licata, Mona Moshet & Andi Deng

2. Table of Contents Tessellations Reflections by Mona Moshet
Dilations by Nicola
Rotation by Yena Shin
Translation by Andi Deng
Bibliography

3. Tessellations

4. Vocabulary Tessellation- a repeating pattern of figures that
Completely covers a plane without any gaps or overlaps
Edge- the intersection between two bordering tiles
Vertex- the intersection of three or more bordering tiles
Regular tessellation- when a tessellation uses only one type of regular polygon to fill up a plane
Semi-regular tessellation- when a tessellation uses more than one type of regular polygon to fill up a plane
Translational symmetry-when a translation maps that tessellation onto itself

5. Vocabulary (continued)
Glide reflectional symmetry-when a glide reflection maps the tessellation onto itself
Reflection or line symmetry-when a figure is reflected across the axis & the image is the same as the original
Rotational symmetry-when a rotation of 180 degrees or less is performed on a tessellation and the resulting image is the same as the original image
Point symmetry-when a tessellation rotates 180 degrees and the image is the same

6. Determining which polygons to tessellate or not
To see which regular polygons can tessellate or not you use the following formula to find the polygon’s angle measure
a=180(n-2)/n
If the angle measure is a factor of 360, then that polygon can tessellate
For example, if you wanted to
see if a regular pentagon, like the one
at the right, could tessellate, you use
the formula to find that each inner
angle is 108 degrees
Then you would divide 360 By 108
360/108=3 1/3, therefore a regular pentagon cannot tessellate

7. Tessellations Activity
Johnson wants to tile the floor of his house. However he wants to semi-tessellate the tiles with 3 different shapes. Can you help Johnson pick which shapes he needs?

8. Reflections
By Mona Moshet

9. Vocabulary
Reflection – a transformation which uses a line that acts like a mirror, with an image reflected in the line
Line of Reflection – the line which acts like a mirror in a reflection
Line of symmetry – the imaginary line where you could fold the image and have both halves match exactly
Isometry – a transformation that preserves length, after the shape is reflected, its corresponding sides remain congruent
Line of symmetry

10. Reflecting Figures & Finding the Line of Reflection
Rules for reflecting figures:
rx-axis (x,y)=(x,-y)
ry-axis (x,y)=(-x,y)
ry=x (x,y)=(y,x)
ry=-x (x,y)=(-y,-x)
ry=n (x,y)=(x,2[n-y]+y)
rx=n (x,y)=(2[n-x]+x,y)
When finding the line of reflection one
could use the midpoint formula. The
midpoint of each pair of corresponding points
is a point on the line of reflection.
In the problem to the right, where is the line
of reflection?

11. Finding the Minimum Distance
To find the minimum distance one must first reflect point A over the given
line.
Then a segment between
A’ & B should be drawn.
The point where segment A’B
intersects the line of
reflection is point C.
Then a segment from A to C
should be drawn
Segment AC is congruent
to segment A’C
In this problem, the mayor of Haemmerle Ville wants a new library built that has the minimum distance from Rite Aid, the town’s beloved pharmacy & the school playground.
In this problem, the mayor of Haemmerle Ville wants a new library built that has the minimum distance from Rite Aid, the town’s beloved pharmacy & the school playground.

12. Activity
Take the paper that you have and bring the top left tip to the right side of the paper till a triangle is made.
Then fold the small rectangle under the triangle. You now have a square!
Fold the square in different ways to find how many lines of symmetry the square has
How many lines of symmetry does a square have?
This proves that a regular polygon’s number of lines of symmetry is equal to it’s number of sides

13. Dilations
Nicola Licata

14. What is a Dilation?
A dilation is a transformation that reduces or enlarges a polygon by a given scale factor around a given center point.
When a figure is dilated the new figure will always be similar to the original figure
Center Point
There is a
scale factor
of 2 which
means that
the new figure
is 2 times larger

15. The Scale Factor
The scale factor is the amount by which the image grows or shrinks
To find the scale factor of two given shapes simply find two corresponding sides or points and put the new shape’s side over the original shape’s corresponding side. 15
Here the scale factor is 3 because 15 is the new distance
and 5 is the old distance. 15/5 = 3 5

16. Reductions and Enlargements
If the scale factor of your dilation is greater than 1 than the dilation is an enlargement.
If your scale factor is greater than 0 but less than 1 than the dilation is a reduction
This is an enlargement because the scale factor is 5.
100 20

17. Activity
Johnny wants to make a scale model of the clock tower big ben in London. If he wants to make it 1/100 of the actual height how tall will the scale model be?
316 ft.

18. Rotation

19. Key Vocabulary Rotation
A transformation where a figure is turned around a center of rotation.
Isometry
A transformation where the figure stays congruent
Center of Rotation
A fixed point anywhere that the figure rotates about. The center of rotation can be anywhere. It can be inside or outside the figure.
Angle of Rotation
The angle created by rays drawn from the center of rotation to a point and its image.

20. Key Concepts
Theorem Line K and line M intersect at point P. Then a reflection in line K and the line M is a rotation about point P. The angle of rotation is double the angle formed by K and M. Since the angle formed by lines Kand M is 70°, the angle of rotation is 140°.

21. Key Concepts Cont. Rotational Symmetry Equations:
A figure in the plane has rotational symmetry if the figure can be mapped onto itself by clockwise rotation of 180 degrees or less.
Equations:
R90 (X,Y) = (-Y,X)
R180 (x,y) =( -x,-y)
R270 (x,y) =(y,-x)
R-90 (x,y) = (y, -x)

22. Rotational Symmetry
Does this figure have rotational symmetry? If so, what is the angle of rotation?
The building has 36 sides so it has a rotational symmetry of 10°

23. Rotating on a Coordinate Plane
You can rotate the building by using the equations from the previous slide. You are to rotate counter clock wise unless told by the problem.
Equations:
R90 (X,Y) = (-Y,X)
R180 (x,y) =( -x,-y)
R270 (x,y) =(y,-x)
R-90 (x,y) = (y, -x)

24. Real- Life Application
The Pentagon has a rotational symmetry of 72° When it is rotated at 72°, it maps onto itself.

25. Save the Leaning Tower of Pisa!
It is 3025 and the leaning tower of Pisa is now leaning too much. Now, the people decide that there is a need to reconstruct the tower. Find the angle you need to rotate the leaning tower to make the tower perpendicular to the ground by using a protractor.

26. Translation
By Andi Deng

27. Vocab Vector --A quantity having direction as well as magnitude
Initial point – starting point of the vector
Terminal point – ending point of the vector
Component form/ coordinate vector– horizontal and vertical values  < a, b >
Coordinate notation– (x,y)  (x+a, y+b)

28. Andi’s explanation It’s pretty much a thing move to another place.
See example: A house moved to a park from a city by a witch. ( not in scale)
Vector component form/ coordinate vector: (15, 3)
Coordinate notation– (x,y)  (x+15, y+3) 3 15

29. Relating to the World
Mobile house
A moving
house
Another
moving

30. practice Make the house transfer 10 boxes to right and 5 boxes up.
Vector component form/
coordinate vector: (10, 5)
Coordinate notation
– (x,y)  (x+10, y+5)
practice
Make the house transfer 10 boxes to right and 5 boxes up.

31. Bibliography http://www.mathnstuff.com/gif/7x7not.gif

32. Bibliography