Transformation Geometry Dilations

What is a Dilation?  Dilation is a transformation that produces a figure similar to the original by proportionally shrinking or stretching the figure. Dilated PowerPoint Slide

Proportionally  When a figure is dilated, it must be proportionally larger or smaller than the original.  Same shape, Different scale. Let’s take a look… We have a circle with a certain diameter. Decreasing the size of the circle decreases the diameter. And, of course, increasing the circle increases the diameter. So, we always have a circle with a certain diameter. We are just changing the size or scale.

Which of these are dilations?? A C D B HINT: SAME SHAPE, DIFFERENT SIZE

Scale Factor and Center of Dilation When we describe dilations we use the terms scale factor and center of dilation.  Scale factor  Center of Dilation Here we have Igor. He is 3 feet tall and the greatest width across his body is 2 feet. He wishes he were 6 feet tall with a width of 4 feet. He wishes he were larger by a scale factor of 2. His center of dilation would be where the length and greatest width of his body intersect.

Scale Factor  If the scale factor is larger than 1, the figure is enlarged.  If the scale factor is between 1 and 0, the figure is reduced in size. Scale factor > 1 0 < Scale Factor < 1

Are the following enlarged or reduced?? A C D B Scale factor of 0.75 Scale factor of 3 Scale factor of 1/5 Scale factor of 1.5

The Object and the Image A’ A B B’ C’ C  The original figure is called the pre-image and the new figure is called the image.  The object is labeled with letters.  The image may be labeled with the same letters followed by the prime symbol. Pre-Image Image

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation used to create an image larger than the original is called an enlargement. A dilation used to create an image smaller than the original is called a reduction.

Dilations always involve a change in size. Notice how EVERY coordinate of the original triangle has been multiplied by the scale factor (x2).

REVIEW: Answer each question……………………….. Does this picture show a translation, rotation, dilation, or reflection? How do you know? Rotation

Does this picture show a translation, rotation, dilation, or reflection? How do you know? Dilation

Does this picture show a translation, rotation, dilation, or reflection? How do you know? (Line) Reflection

DILATION 200%50% Notice each time the shape transforms the shape stays the same and only the size changes. ENLARGEREDUCE

Look at the pictures below DILATION Dilate the image with a scale factor of 75% Dilate the image with a scale factor of 150%

Look at the pictures below DILATION Dilate the image with a scale factor of 100% Why is a dilation of 75% smaller, a dilation of 150% bigger, and a dilation of 100% the same?

Example 3 Dilations in the Coordinate Plane COORDINATE GEOMETRY Triangle ABC has vertices A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after a dilation centered at the origin with the scale factor of 2. Sketch the preimage and the image

Example 3 Dilations in the Coordinate Plane COORDINATE GEOMETRY Triangle ABC has vertices A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after a dilation centered at the origin with the scale factor of 2. Sketch the preimage and the image Preimage (x, y) Image (2x, 2y) A(7, 10)A’(14, 20) B(4, -6)B’(8, -12) C(-2, 3)C’(-4, 6)

Example 4 Identify the Scale Factor Determine the scale factor for each dilation with center C. Then determine whether the dilation is an enlargement, reduction, or congruence transformation. a D E B A’ C A B’ E’ D’ Scale factor = image length preimage length = 6 units 3 units = 2

1.  ABC has coordinates A(-2,1), B(1,4) and C(2,2) a. State the coordinates of  A’B’C’, the image of  ABC, after D 2.  ABC after D2. A’ (-4, 2) B’ (2, 8) C’ (4, 4)  ABC after D½ A” (-1, ½) B’’ (½, 2) C’’ (1, 1)  ABC A (-2, 1) B (1, 4) C (2, 2) 1.  ABC has coordinates A(-2,1), B(1,4) and C(2,2) b. State the coordinates of  A”B”C”, the image of  ABC, after D ½  ABC A (-2, 1) B (1, 4) C (2, 2)

Find the length of the sides of the dilated objects based on the scale factor: Pre-Image Triangle Side 1Side 2Side 3 9 inches13 inches8 inches \Dilate the sides of the original triangle by a scale factor of 3: Image Triangle Side 1Side 2 Side 3 27 inches39 inches24 inches Dilations change the size of the shape. Each length is reduced or enlarged by the scale factor. If the center of dilation is not at the origin then you have to use the measures of the sides as references.

Dilation Dilate the following figures on the given coordinate planes: Center of Dilation = Point S Scale Factor = 2 ST U V S´S´ U´U´ V´V´ T´T´ 7 7(2) = (2) = (2) = (2) = 8 8

Center of Dilation is Point Y Scale Factor is Y K P L C E K´K´ P´P´ L´L´ C´C´ E´E´

 PROBLEM: Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2.  OBSERVE: Point E and its image are the same. It is important to observe the distance from the center of the dilation, E, to the other points of the figure. Notice EF = 6 and E'F' = 3.  HINT: Be sure to measure distances for this problem.

Dilations Used Everyday

The End

Remember  Dilations are enlargements or reductions.  What are some things that you would not mind dilating to make larger or smaller? Practice Dilation Quiz

If is > 1, the image is an enlargement of the original figure. If 0 < <1, the image is a reduction of the original figure. If = 1, the image is congruent to the original figure. Characteristics of Dilations

Example 4 Identify the Scale Factor Determine the scale factor for each dilation with center C. Then determine whether the dilation is an enlargement, reduction, or congruence transformation. a J F H C G Scale factor = image length preimage length = 4 units 4 units = 1 The dilation is a congruence transformation