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Transformation PPT-Translations, Rotation, Reflection and Dilation.pptx

This document provides a comprehensive overview of geometric transformations, including translation, reflection, rotation, and dilation. It details each transformation's characteristics, properties, and examples, highlighting their effects on shape, size, and orientation. The document also discusses congruent and similar figures, and includes practice exercises for applying these transformations in the coordinate plane.

Education◦
TRANSFORMATIONS IN THE COORDINATE PLANE
IN THIS LESSON, WE WILL: • Review the four geometric transformations • Identify unique geometric properties of each geometric transformation • Determine which geometric transformations produce congruent figures • Identify which geometric transformations produce similar figures • Practice transforming figures in the coordinate plane • Use appropriate notation to represent all geometric transformations
TRANSLATION • Moves an object up, down, left, and/or right in the coordinate plane • Slides an object around • Preserves shape AND size • All distances are the same • Creates congruent figures
TRANSLATION EXAMPLE • Notice that the pre-image and the image are congruent: same shape and same size • Notice that the orientation of the pre- image and the image are the same • A special property of translations is that the distances between corresponding vertices on the pre-image and the image are equal
TRANSLATION A translation is a transformation that slides a figure across a plane or through space. With translation all points of a figure move the same distance and the same direction.
TRANSLATION Basically, translation means that a figure has moved. An easy way to remember what translation means is to remember… A TRANSLATION IS A CHANGE IN LOCATION. A translation is usually specified by a direction and a distance.
TRANSLATION What does a translation look like? A TRANSLATION IS A CHANGE IN LOCATION. x y Translate from x to y original image
TRANSLATION In the example below triangle A is translated to become triangle B. A B Describe the translation. Triangle A is slide directly to the right.
TRANSLATION In the example below arrow A is translated to become arrow B. Describe the translation. Arrow A is slide down and to the right. A B
REFLECTION • Flips an object across a line of reflection in the coordinate plane • Changes an object’s orientation (direction) • Preserves shape AND size • Creates congruent figures
REFLECTION EXAMPLE • Notice that the pre-image and the image are congruent: same shape and same size • Notice that the orientations of the pre- image and the image are different • A special property of reflections is that the segment from any vertex on the pre-image to its corresponding vertex on the image is a perpendicular bisector of the line of reflection
REFLECTION A REFLECTION IS FLIPPED OVER A LINE. A reflection is a transformation that flips a figure across a line.
REFLECTION A REFLECTION IS FLIPPED OVER A LINE. After a shape is reflected, it looks like a mirror image of itself. Remember, it is the same, but it is backwards
REFLECTION The line that a shape is flipped over is called a line of reflection. A REFLECTION IS FLIPPED OVER A LINE. Line of reflection Notice, the shapes are exactly the same distance from the line of reflection on both sides. The line of reflection can be on the shape or it can be outside the shape.
REFLECTION Determine if each set of figures shows a reflection or a translation. A REFLECTION IS FLIPPED OVER A LINE. A B C A’ B’ C’
REFLECTION Sometimes, a figure has reflectional symmetry. This means that it can be folded along a line of reflection within itself so that the two halves of the figure match exactly, point by point. Basically, if you can fold a shape in half and it matches up exactly, it has reflectional symmetry.
REFLECTIONAL SYMMETRY An easy way to understand reflectional symmetry is to think about folding. Do you remember folding a piece of paper, drawing half of a heart, and then cutting it out? What happens when you unfold the piece of paper?
REFLECTIONAL SYMMETRY The two halves make a whole heart. The two halves are exactly the same… They are symmetrical. Reflectional Symmetry means that a shape can be folded along a line of reflection so the two haves of the figure match exactly, point by point. The line of reflection in a figure with reflectional symmetry is called a line of symmetry. Line of Symmetry
REFLECTIONAL SYMMETRY The line created by the fold is the line of symmetry. A shape can have more than one line of symmetry. Where is the line of symmetry for this shape? How can I fold this shape so that it matches exactly? NOT THIS WAY I CAN THIS WAY Line of Symmetry
REFLECTIONAL SYMMETRY How many lines of symmetry does each shape have? Do you see a pattern?
REFLECTIONAL SYMMETRY Which of these flags have reflectional symmetry? United States of America Mexico Canada England
ROTATION • Spins an object about a fixed point in the coordinate plane • Usually changes an object’s orientation (direction) • Preserves shape AND size • Creates congruent figures
ROTATION EXAMPLE • Notice that the pre-image and the image are congruent: same shape and same size • Notice that the orientations of the pre- image and the image are different, unless you rotate one 360 degrees
A SPECIAL PROPERTY OF ROTATIONS • A point on the pre-image and its corresponding point on the image lie on a circle whose center is the center of rotation.
ANOTHER SPECIAL PROPERTY OF ROTATIONS • Line segments connecting corresponding points on the pre-image and image to the center of rotation are congruent and form an angle whose measure equals that of the angle of rotation.
ROTATION A rotation is a transformation that turns a figure about (around) a point or a line. The point a figure turns around is called the center of rotation. Basically, rotation means to spin a shape. The center of rotation can be on or outside the shape.
ROTATION What does a rotation look like? A ROTATION MEANS TO TURN A FIGURE center of rotation
ROTATION This is another way rotation looks A ROTATION MEANS TO TURN A FIGURE The triangle was rotated around the point. center of rotation
ROTATION If a shape spins 360, how far does it spin? All the way around This is called one full turn. 360
ROTATION If a shape spins 180, how far does it spin? Half of the way around This is called a ½ turn. 180 Rotating a shape 180 turns a shape upside down.
ROTATION If a shape spins 90, how far does it spin? One-quarter of the way around This is called a ¼ turn. 90
ROTATION Describe how the triangle A was transformed to make triangle B A B Describe the translation. Triangle A was rotated right 90
ROTATION Describe how the arrow A was transformed to make arrow B Describe the translation. Arrow A was rotated right 180 A B
ROTATION When some shapes are rotated they create a special situation called rotational symmetry. to spin a shape the exact same
ROTATIONAL SYMMETRY A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape. Here is an example… As this shape is rotated 360, is it ever the same before the shape returns to its original direction? Yes, when it is rotated 90 it is the same as it was in the beginning. So this shape is said to have rotational symmetry. 90 
ROTATIONAL SYMMETRY Here is another example… As this shape is rotated 360, is it ever the same before the shape returns to its original direction? Yes, when it is rotated 180 it is the same as it was in the beginning. So this shape is said to have rotational symmetry. 180  A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.
ROTATIONAL SYMMETRY Here is another example… As this shape is rotated 360, is it ever the same before the shape returns to its original direction? No, when it is rotated 360 it is never the same. So this shape does NOT have rotational symmetry. A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.
ROTATION SYMMETRY Does this shape have rotational symmetry? 120  Yes, when the shape is rotated 120 it is the same. Since 120  is less than 360, this shape HAS rotational symmetry
DILATION • An enlargement or reduction of a figure by a scale factor in the coordinate plane • Changes the size of an object • Preserves shape • Creates similar figures
DILATION EXAMPLE • Notice that the pre-image and the image are similar: corresponding angles are congruent and corresponding sides are proportional • Notice that the orientations of the pre- image and the image are the same • A special property of dilated figures is that you can compute the scale factor of the dilation by finding the ratio
CONCLUSION We just discussed three types of transformations. See if you can match the action with the appropriate transformation. FLIP SLIDE TURN REFLECTION TRANSLATION ROTATION
Translation, Rotation, and Reflection all change the position of a shape, while the size remains the same. The fourth transformation that we are going to discuss is called dilation.
Dilation changes the size of the shape without changing the shape. DILATION When you go to the eye doctor, they dilate you eyes. Let’s try it by turning off the lights. When you enlarge a photograph or use a copy machine to reduce a map, you are making dilations.
Enlarge means to make a shape bigger. DILATION Reduce means to make a shape smaller. The scale factor tells you how much something is enlarged or reduced.
DILATION 200% 50% Notice each time the shape transforms the shape stays the same and only the size changes. ENLARGE REDUCE
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?
Lets try to make sense of all of this TRANSFORMATIO NS CHANGE THE POSTION OF A SHAPE CHANGE THE SIZE OF A SHAPE TRANSLATI ON ROTATION REFLECTIO N Change in location Turn around a point Flip over a line DILATION Change size of a shape
See if you can identify the transformation that created the new shapes TRANSLATION
See if you can identify the transformation that created the new shapes REFLECTION Where is the line of reflection?
See if you can identify the transformation that created the new shapes DILATION
See if you can identify the transformation that created the new shapes ROTATION
See if you can identify the transformation in these pictures? REFLECTION
See if you can identify the transformation in these pictures? ROTATION
See if you can identify the transformation in these pictures? TRANSLATION
See if you can identify the transformation in these pictures? DILATION
See if you can identify the transformation in these pictures? REFLECTION
SUMMARY PRESERVES SHAPE • Translation • Reflection • Rotation • Dilation PRESERVES SIZE • Translation • Reflection • Rotation
SUMMARY PRESERVES ORIENTATION • Translation • Dilation
CONGRUENT VS. SIMILAR FIGURES PRODUCES CONGRUENT FIGURES • Translation • Reflection • Rotation PRODUCES SIMILAR FIGURES • Dilation
PRACTICE USING TRANSFORMATIONS AND THEIR PROPERTIES Work through this example in your notes.
1. TRANSFORM TRIANGLE ABC BY MOVING IT THREE UNITS LEFT AND TWO UNITS UP. THEN, DILATE IT BY A FACTOR OF ½. • Notice the coordinates: • A(2, 2) • B(-2, 3) • C(-5, 0) • We will do the translation first. We have to move the triangle left three units and up two units. This means f(x, y) = (x – 3, y + 2).
1. TRANSFORM TRIANGLE ABC BY MOVING IT THREE UNITS LEFT AND TWO UNITS UP. THEN, DILATE IT BY A FACTOR OF ½. • Notice the new coordinates after we subtract 3 from each x and add 2 to each y: • A(-1, 4) • B(-5, 5) • C(-8, 2) • Next, let’s dilate the triangle by a factor of ½. This would be represented by f(x, y) = (1/2x, 1/2y).
1. TRANSFORM TRIANGLE ABC BY MOVING IT THREE UNITS LEFT AND TWO UNITS UP. THEN, DILATE IT BY A FACTOR OF ½. • Now we have our new triangle: • A’ (-1/2, 2) • B’ (-5/2, 5/2) • C’ (-4, 1) • We put the two changes together to write a function rule that takes us directly from the pre-image to the image. This would be represented by f(x, y) = (1/2(x - 3), 1/2(y + 2), or f(x, y) = (1/2x – 3/2, 1/2y + 1)