Unit 1 Transformations Congruence and Similarity Basic Types

Unit 1: Transformations, Congruence, and Similarity

Basic Types of Transformations: Translations Reflections Rotations

Quadrant II x-axis Quadrant III Quadrant I (x, y) Quadrant IV y-axis

Object to Image (Before) (After) Before Transformation: After Transformation: (‘ = PRIME) A’ A B C B’ C’

Translations…

A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. Objects that are translated are congruent. *The word "translate" in Latin means "carried across".

Example 1: Translate the object down 2 and right 3 units.

Example 1 Solution: Translate the object down 2 and right 3 units.

Example 2: Translate the object (-3, 4)

Example 2 Solution: Translate the object (-3, 4)

Remember: Translations are SLIDING on a graph!!! The shape doesn’t change at Translations are SLIDES!!! all.

Reflections…

A reflection “flips” an object and can be seen in water, in a mirror, in glass, or in a shiny surface. An object and its reflection have the same shape and size, but the figures face in opposite directions. In a mirror, for example, right and left are switched.

The line (where a mirror may be placed) is called the line of reflection. The distance from a point to the line of reflection is the same as the distance from the point's image to the line of reflection. A reflection can be thought of as a "flipping" of an object over the line of reflection. The object ABCD is being reflected over the x-axis.

Example 3: Reflect the object over the y-axis.

Example 3 Solution: Reflect the object over the y-axis.

Example 4: Reflect the object over x = 2.

Example 4 Solution: Reflect the object over x = 2.

Rotations…

A rotation is a transformation that turns a figure about a fixed point called the center of rotation. An object and its rotation are the same shape and size, but the figures may be turned in different directions.

Rotations Graphically… Physically rotate the graph paper and use the original points – just graphed in the different quadrants.

Rules: When students complete rotating the original figure clockwise 90˚, 180˚, and 270˚, you can have them come up with the rules on their own. If you have time, you can have them use the same figure and rotate counter clockwise and come up with the rules of those too.

Rotation Rules: Clockwise Counter Clockwise 90˚ (y, -x) (-y, x) 180˚ (-x, -y) 270˚ (-y, x) (y, -x)

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 Power. Point Slide

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

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

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. His center of dilation would be where the He wishes he were larger by a length and greatest width of his body scale factor of 2. intersect.

The Object and the Image B’ The original figure is called the object 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. Image C’ A’ B Object A C

Determining Scale Factor:

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

Ratio Fraction Decimal Percentage 1: 2 2: 5 1/2 3/4 1/8 . 5 0. 9 50% 400% Reduction or Enlargement Reduction

Are the following enlarged or reduced? ? C A Scale factor of 1. 5 D B Scale factor of 0. 75 Scale factor of 1/5 Scale factor of 3

Dilations Used Everyday

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

Dilation transformation that changes the size of an object, but not the shape. A Dilation will be a similar figure, but not a congruent figure. A Example:

Dilate the object by a scale factor of ½ (-2, 2) (2, -2) (-2, -2)

Dilate the object by a scale factor of 3 (-6, 6) (-2, 2) (2, -2) (-2, -2) (6, -6) (-6, -6)

A spider has taken up residence in a small cardboard box which measures 2 inches by 4 inches. What is the length, in inches, of a straight spider web that will carry the spider from the lower right front corner of the box to the upper left back corner of the box? A. 4. 47 in. B. 5. 66 in. C. 5 in. D. 6 in.