Transformations at the Beach . By Daniella Kay, Lucy Almberg, Greg Lubin, and James Kim . Table Of Contents . Reflections- Slide 3 Dilations- Slide 13 Translations- Slide 19 Tessellations- Slide 24 Rotations- Slide 26. Reflections .
By Daniella Kay, Lucy Almberg, Greg Lubin, and James Kim
a transformation which uses a line that acts as a mirror to reflect an image in that line.
Beach towels are Sunglasses
rectangles and have 1 line of
have 2 lines of symmetry symmetry
Johnny owns a surf shop on an island with two beaches. He wants his surf shop to be an equal distance from each beach. Use minimum distance to help Johnny decide where on the road (line m) he should put his shop.
A dilation with the center C and scale factor K is a transformation that maps every point P in the plane to a point P’ so that the following properties are true.
1. If P is not the center point C, then the image P’ lies on CP. The scale factor k is a positive number such that k=
and k =1
2. If P is the center point C, then P = P’
Scale Factor= 2
Translation - A transformation that maps every two points so that the preimage and image of the segment made by connecting them for both points are parallel, collinear and congruent
Initial point- the starting point of the the vector of a translation
Terminal point- the ending point of a vector at the image
Component form - A form of a vector that combines the horizontal and vertical components
A vector can be used to describe a translation by mapping out the distance of points from the starting point to the terminal point with horizontal and vertical distance.
So if point A was (1,4) and the image of point A was (3,6) the vector form of the translation would be <2,2> because the point moved two units to the right on the x-axis and 2 units up on the y-axis.
Coordinate notation is another way of describing a translation. For example, if we use point A (1,4) and point A’ (3,6), the way to describe that translation with coordinate notation would be (x + 2, y+2).
Matrices are a different way to describe transformations, points, and shapes in a coordinate plane.
For point A (4,5), in matrix form it would be [ 4 ] for describing a translation [ 2 ]. 5
The matrix for the translation means two units right and two units up. To figure out the image of the translation you would add them.
So [ 4 ] + [ 2 ] would be [ 6 ]
5 2 7
All translations are isometries, that means that the preimage and the image of the polygon, or segment that is translated will be congruent.
Not a Translation
Is a Translation
The segments that are made when connecting the preimage to the image for the points in a translation are always parallel and congruent to each other.
In the real world, vectors and coordinate notation are used to describe movement with coordinates.
The boat at the shore was at 39 degrees and 55 minutes North and 74 degrees and 4 minutes West (-74.07, 39.92) and the boat at the ocean was at
39 degrees and 53 minutes North and 74 and 3 minutes West (-74.05, 39.88)
Can you write in vector form and coordinate notation the translation?
Frieze pattern (or border pattern) - a pattern that extends to the left and right in a way so that a pattern can be mapped onto itself with horizontal translation.
Types of Frieze patterns:
TR: Translation and 180 degree rotation
TG Translation and horizontal glide reflection
TV: Translation and vertical line reflection
TRVG: Translation, 180 degree rotation, vertical line reflection and horizontal glide reflection
TRHVG: Translation, 180 degree rotation, horizontal glide reflection, vertical line reflection
A frieze pattern with horizontal translation:
A frieze pattern with horizontal translation and 180 degree rotation:
A rotation is a transformation in which a figure is turned about a fixed point
The starfish has 5 sides
The angle of rotation of the starfish is 72º.
Rotate ABC 140° counter clockwise about point K.
2. Measure a 140° angle using KA as one side and mark it.
3. Measure the length of KA.
4. Draw a segment the same length as KA, going from point K towards the 140° mark and make that point A’.
5. Erase all the lines you drew but keep point A’.
6. Repeat steps 1-5 for vertices B and C and then draw
Just use these equations to rotate the figure:
Clockwise: Counter Clockwise:
R90- (x,y) = (y, -x) R90- (x,y) = (-y, x)
R180- (x,y) = (-x, -y) R180- (x,y) = (-x, -y)
R270- (x,y) = (-y, x) R270- (x,y) = (y,-x)
Rotate the figure 180º clockwise.
Coordinates of Coordinates of
the Preimage: the Image:
C ( 5,-1)
D (2, -1)
What is the angle of rotation?
N V R
S O O
F C S
O A S
R B W
"Beach Arrow Facebook Covers for Timeline." Addcovers.com. N.p., n.d. Web. 6 May
"Ocean Shell." Photo-dictionary.com. N.p., n.d. Web. 6 May 2014.