Around the World with Transformations

Table of Contents

Rotations: Slides 3-13 (Jake Campbell)Translations: Slides 14-19 (Alison Sorkenn)Dilations: Slides 20-29 (Eric Park)Reflections: Slides 30-40 (Angela Hong)Tessellations: Slides 41-47 (Conor McCorry)

Rotations

Rotations● A rotation is when a figure is turned about a fixed point● Rotations are isometries (Segment and angle measures remain congruent)● This point is called the center of rotation● The amount of degrees that the figure is rotated about that point is the angle

of rotation ● Unless noted, the angle of rotation is always counter-clockwise

Rotations on the Coordinate Plane● To rotate points counter-

clockwise around the origin when given coordinates, use the equations below

● 90°- (x,y) = (-y,x)● 180°- (x,y) = (-x,-y)● 270° - (x,y) = (y,-x)● For clockwise rotations the

equations are● 90°- (x,y) = (y,-x)● 180°- (x,y) = (-x,-y)● 270° - (x,y) = (-y,x)

Activity:Rotate ΔABC 180° about the originA= (1,1)B= (4,1)C= (5,3)

A′=B′=C′=Now plot the points of the image

and preimage.

(-1,-1)

(-4,-1)(-5,-3)

A B

C

A’B’

C’

Finding Angles of Rotation

● Reflecting over two intersecting lines is the same as rotating

● To find the angle of rotation, find the angle of where the lines meet and multiply it by 2

32°

In image on the left, what is the angle of rotation?

Answer: 64°

Rotational Symmetry● If a figure can be rotated onto itself in 180° or less, it has rotational symmetry● All regular polygons have rotational symmetry● You can find the angle of rotation for regular polygons by by using the formula

360/n (number of sides)● To determine if an irregular shape has rotational symmetry, you have to eyeball it● If the irregular shape has rotational symmetry, figure out how many ways the

shape can map on to itself with rotations less than or equal to 360°● The number of times a shape can map onto itself is known as the order● To find the angle of rotational symmetry, use the equation 360/o (order)

Real-Life Examples of Rotational Symmetry

The flag of Bangladesh has 180° rotational symmetry

This Hindu (the primary religion of Bangladesh) swastika has 90° rotational symmetry

Does this flower, native to Bangladesh, have rotational symmetry?If so, what is the angle of rotation? (Assume all petals are congruent)

Answer: The flower has 72° rotational symmetry. Since it can be rotated onto itself 5 times within a 360° rotation, its order is 5. When we use the equation 360/o, we get 72°.

Rotating Around a Point Using a Ruler and Protractor (Original Artwork)

1. Rotate triangle ABC 90° clockwise around point D

2. Draw a line segment from point C to the point of rotation (P) 3. Using a protractor, create the

angle of rotation (90°) using point P as the vertex. Make sure that the angle is in the direction that you’re rotating (clockwise)

4. Finish drawing in the angle

5. Measure the length of CP 6. Construct point C’ the length that you previously obtained along the angle that you drew in.

7. Erase angle CPC’ and repeat steps 1-6 for point A

8. Repeat steps 1-6 for point B. The final, rotated triangle should look like this.

Translations

•Translation is a transformation that maps original points to their final points

•Vector is a quantity that has both direction and magnitude

•A translation is an isometry, which means that the lengths stay the same.

<6,-2> 6 units right, 2 units down

•When an object is translated, all points are moved the same distance and in the same direction.

Key Words to Know

● Reflecting an object twice over parallel lines results in a translation.

Translating Objects•When an object is translated, it can be moved to any quadrant, as long as it stays in the same direction and is the same size.

•Original objects are known as pre-images and translated objects are the images of the original.

y

x

Rules to Use for Translations•There are three different ways to write out translations.

•Coordinate Notation: -represents how the coordinates are translated; adding/subtracting x and y

•Vector in Component Form: <6,-2>-shows movement of the coordinates

•Matrices:

-rows and columns showing x and y coordinates

-use matrix addition to find translated points

xy

(x,y) → (x+a,y+b)

Finding Image of Pre-Image•If you are given a pre-image with points on

a graph and a rule, you will have to figure out the points of the image.

•Ex: A(-8,7) B(-8,1) C(-1,7) D(-1,1)

•Rule= (x,y) → (x+10,y+2)

•Find the coordinates of the image and the vector that describes the new image.

•The x-coordinate will increase by 10 units and the y-coordinate will increase by 2 units.

● Final Points: A(2,9) B(2,3) C(9,9) D(9,3)

Pre-Image Image

A C

B D

A’ C’

B’ D’

Word Problem•The Brazil soccer team is playing a game and needs to score a goal. One of the players with the ball sees an open player and decides to pass it to him. Write the vector in component form that describes the translating of the ball from player to player.

Dilations

DilationsDilation- with a center of C and a scale factor of K, is a transformation that maps every point from the

pre-image to their corresponding points of the dilated image1. If p is not the center point C, then the image point P’ lies on CP. The scale factor k is a positive

number such that k = CP’/CP, and K =/=12. If P is the center point C, then P = P’Reduction- A dilation is a reduction if: 0 < K < 1Enlargement- A dilation is an enlargement if: K > 1

Finding the scale factor of an image

Put the value of the dilated image over the value of the preimage

Preimage5 inches

Dilated image10 inches

Dilated image/ Preimage10/5 (Then reduce the fraction)2/1The scale factor of this dilation is 2

Dilating Images with the origin as the center

Matrix- rectangular arrangement of numbers in rows and columnsScalar multiplication- multiplying a matrix by a real number Multiply the x and y coordinates of the primage by the scale factor to get the coordinates of the dilated imageEx:(Scale factor 3)Coordinates of Preimage: (0,4) (5,-1) (-2,2)

X [ 0    5   -2 ]  [ (3)0   (3)5    (3)-2 ] [ 0  15 -6] Y [4     -1    2   ]     [ (3)4   (3)-1   (3)2   ] [ 12 -3  6]

Coordinates of Dilated image: (0,12)(15,-3)(-6,6)

Dilating Images with the origin as the center

Use scalar multiplication to find the coordinates of the dilated imageGiven: Scale factor = 2. (0,4) (2,3) (3,1)

Answer:[ (2)0 (2)2 (2)3] [ (2)4 (2)3 (2)1]

[ 0 4 6] (0,8) (4,6) (6,2)[ 8 6 2]

Dilating Images with the origin as the center

Plot the points on the graph from the previous slide:

Dilating Images with the origin as the centerNow you try! Dilate the coordinates (1,3) (2,4) (5,1). (First graph these points). With a scale factor of 3 and

the center as the origin

A’=(3,9)B’=(6,12)C’=(15,3)

Answers:

Dilating Images with the origin as the center

The British soccer team Manchester United uses three people to pass the ball between. They each travel back the same distance as the others. If the original coordinates of the three people are: (1,4)(3,6)(4,2). And the coordinates of the three people after traveling back an equal amount: (4,16)(12,24)(16,8). Find the scale factor.

Answer:Dilated coordinate over preimage coordinate: 4/1= 4 is the scale factor

Challenge yourselfImagine you are a toy modeler. You are trying to make a replica of the building Big Ben. Big Ben is approximately 316 feet. The height of your model is 66 inches. The width of Big

Ben is 50 feet. Find the width of your model.

Answer:

First convert feet to inches (You want your toy model in inches). (3792 inches, and 600 inches for width)Then put the dilated image over the preimage. 66/3792 = 0.017 which is your scale factor.Then multiply 50 by your scale factor.

(which gets you 0.87inches) Which is the answer

Dilating an image with a center that is not the origin

When the center of the dilation is not the origin, you can’t simply multiply the x and y coordinates of the preimage by the scale factor. Use this equation when dilating an image on a coordinate plane. (This equation does work when the center is the origin, but that just adds extra work)

K(x-a)+a K(y-b)+b Where: (x,y) are the coordinates of a point on a preimage (a,b) coordinates of the center of the dilation K is the scale factor

Reflections

Reflections● A reflection is a transformation in which an image is reflected over a line● The line of reflection is the line over which the image is reflected● A figure has a line of symmetry if it can be mapped onto itself by a

reflection over the line○ Reflections are isometries, meaning that the lengths of the preimage

(original) and image (transformed figure) are preserved

An easier way to think about reflections is through the comparison of mirrors.

● The object you hold up to a mirror is the preimage, and its reflection in the mirror is the image

● The line of reflection acts as the mirror itself, because it is what reflects the image onto a new space.

MATH is reflected over the line y = -x

Reflections

Above, the image reflects itself over the line x = 0

The Louvre Pyramid Zis reflected in the water at night

Determining Lines of Symmetry

A figure has a line of symmetry if one side of the shape can be reflected over the line and be congruent to the other. In other words, imagine “folding” the figure over the line, like a piece of paper.

NOThis is not a line of symmetry.

YESThis is a line of symmetry.

Determining Lines of SymmetryTriangles can have 3, 1, or 0 lines of symmetry

In a regular polygon, the number of lines of symmetry is equal to the number of sides the polygon has, because regular polygons have equal sides and angles.

Hexagons can have different lines of symmetry depending on their shape

3 Sides = 3 Lines 4 Sides = 4 Lines 5 Sides = 5 Lines 6 Sides = 6 Lines

Equilateral Isosceles Right RegularIrregular Irregular

The Palace of Versailles is an example of a symmetrical landmark.The line of symmetry runs vertically through the middle of the palace and splits the figure into two congruent halves.

The Eiffel Tower also has a vertical line of symmetry.

1. Pick a vertex on the preimage, and count the number of units it takes to reach the line of reflection by using “rise/run”. Make sure you are counting box-by-box!

2. Count that same number of units from the line of reflection in the opposite direction of the preimage and plot the new point.

3. Repeat steps 1-2 for all the points of the preimage.

4. Connect the plotted points to form the new image.

When reflecting a preimage over a line on a coordinate plane, you can either…

OR

Reflecting a Preimage With Coordinates

Use These Rules:

Reflect over x-axis (x, y) (x, - y)

Reflect over y-axis (x, y) (- x, y)

Reflect over y = x (x, y) (y, x)

Reflect over x = y (x, y) (- y, - x)

Write An Equation for a Line of ReflectionFor lines…

1. Choose a point on the preimage and locate the corresponding point on the image

2. Use midpoint formula to find the midpoint of the two points

3. Find the perpendicular slope of the two points by taking the opposite recipricol of the regular slope

4. Use the perpendicular slope as “m” to create a slope-intercept equation by plugging in the midpoint coordinates as “x” and “y”

For polygons…1. Choose a point on the preimage and

locate the corresponding point on the image

2. Use midpoint formula to find the midpoint of the two points

3. Repeat steps 1-24. Find the slope of the two plotted points

and use it to create an equation by plugging in one of the midpoint coordinates as “x” and “y”

Finding Minimum Distance

Find point C on line m so that AC + BC is of minimum distance.

1. Reflect one of the points over the line

2. Use distance formula or measure with a ruler to find the distance between the reflected and the other unreflected point (A’ and B, or A and B’)

3. The distance calculated is the minimum distance

mBA

BA

A’B

A’

7 cm

Minimum Distance = 7 cm

Now You Try!Jocelyn’s town is installing new water pipes on Chestnut Avenue. Her dad is in charge of the construction project, and asks her to help. The pipes must stretch from House A to House D, but they must also reach the pump of the reservoir on the other side of the street in order to draw water. What is the least amount of piping Jocelyn’s dad can use?

A DCB

RESERVOIRFor this problem:Assume 1 cm = 10 yards

Now You Try: Answer1. The pipes must go from House A to House

D. All other Houses in between are irrelevent to the problem.

2. Reflect one of the houses over the street, and use a ruler to measure the distance between House A’ and House D. Distance formula cannot be used in this particular problem because no coordinates were given.

3. The distance measured is 5 cm, and since 1 cm = 10 yards, the least amount of piping Jocelyn’s dad must use is 50 yards.

DA

A’

A D

Minimum Distance = 5 cm1 cm = 10 yards 5 cm = 50 yards* Not to Scale *

Tessellations

Key Vocabulary

A frieze pattern or border pattern is a pattern that extends to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation.

Tessellations are isometries, because the lengths of the preimage and image are unchanged.

Classification of Frieze PatternT TranslationTR Translation 180 degree rotationTG Translation and horizontal glide reflectionTV Translation and vertical line reflectionTHG Translation, 180 degree rotation, and horizontal glide reflectionTRVG Translation, 180 degree rotation, vertical line reflection, and horizontal glide reflectionTRHVG Translation, 180 degree rotation, horizontal line reflection, vertical line reflection, and horizontal line reflection

Determining what Tessellates Using the equation 180(n-2)/n, you can determine what objects tessellate

n=number of sides If the answer you get is a factor of 360, then the object will tessellate

Determine if a dodecagon tessellates

180(12-2)/12

180(10)/12

1800/12

150

360/150=2.4

This is an example of a tesselation.

Determine the Pattern

1.

2.

TV

T or THG

A B C D

E F

1. Name the Transformation that maps A onto B

2. Name the Transformation that maps B onto E

3. Name the Transformation that maps A onto E

A1. T

A2. TG

A3. THG

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