5.7 Reflections and Symmetry

Objective Identify and use reflections and lines of symmetry.

Key Vocabulary Preimage Image Reflection Line of Symmetry

Identifying Transformations Figures in a plane can be –Reflected –Translated – (did section 3.7) –Rotated – (later) To produce new figures. The new figure is called the IMAGE. The original figure is called the PREIMAGE. The operation that MAPS, or moves the preimage onto the image is called a transformation.

Definition – anything that maps (or moves) an original geometric figure, the preimage, onto a new figure called the image. Transformation

Some facts Some transformations involve labels. When you name an image, take the corresponding point of the preimage and add a prime symbol. For instance, if the preimage is A, then the image is A’, read as “A prime.”

Mapping Preimage Original shape or object. A Image Shape or object after it has been moved. A’

Reflection (flip) A reflection or flip is a transformation over a line called the line of reflection. Each point of the preimage and its image are the same distance from the line of reflection. A A’

Reflection (flip) The result of a figure flipped over a line of reflection.

Click on this trapezoid to see reflection. The result of a figure flipped over a line of reflection.

Reflections Look at yourself in a mirror! –How does your reflection respond as you step toward the mirror? away from the mirror? When a figure is reflected, the image is congruent to the original. The actual figure and its image appear the same distance from the line of reflection, here the mirror.

Example 1: Identifying Reflections Tell whether each transformation appears to be a reflection. Explain. No; the image does not Appear to be flipped. Yes; the image appears to be flipped across a line.. A. B.

Your Turn Tell whether each transformation appears to be a reflection. a. b. No; the figure does not appear to be flipped. Yes; the image appears to be flipped across a line.

Properties

Example 2 Identify Reflections Tell whether the red triangle is the reflection of the blue triangle in line m. SOLUTION Check to see if all three properties of a reflection are met. ∆FGH has a clockwise orientation. ∆F'G'H' has a counter clockwise orientation. 1. Is the image congruent to the original figure? Yes. 2. Is the orientation of the image reversed? Yes. 3.Is m the perpendicular bisector of the segments connecting the corresponding points? Yes.

Example 2 Identify Reflections To check, draw a diagram and connect the corresponding endpoints. Because all three properties are met, the red triangle is the reflection of the blue triangle in line m. ANSWER

Example 3 Identify Reflections The red triangle is not a reflection of the blue triangle. ANSWER Check to see if all three properties of a reflection are met. SOLUTION 1. Is the image congruent to the original figure? Yes. 2. Is the orientation of the image reversed? No. Tell whether the red triangle is the reflection of the blue triangle in line m.

Drawing Reflections Copy the triangle and the line of reflection. Draw the reflection of the triangle across the line. Step 1 Through each vertex draw a line perpendicular to the line of reflection.

Step 2 Measure the distance from each vertex to the line of reflection. Locate the image of each vertex on the opposite side of the line of reflection and the same distance from it. Drawing Reflections Continued

Step 3 Connect the images of the vertices.

1.Identify the y-axis as the line of reflection (the mirror). 2. Point A is 1 unit to the right of the y-axis, so its reflection A’ is 1 unit to the left of the y-axis. We discovered an interesting phenomenon: simply change the sign of the x- coordinate to reflect a point over the y- axis! To graph the reflection of point A(1,-2) over the y-axis: Similarly, to graph the reflection of a point over the x-axis, simply change the sign of the y-coordinate! Reflections in a Coordinate Plane

Example 4 Reflections in a Coordinate Plane SOLUTION a. The x- axis is the perpendicular bisector of AJ and BK, so the reflection of AB in the x- axis is JK. Which segment is the reflection of AB in the x- axis? Which point corresponds to A ? to B ? a. b. Which segment is the reflection of AB in the y- axis? Which point corresponds to A ? to B ? A is reflected onto J. A(–4, 1) J(–4, –1) B is reflected onto K. B(–1, 3) K(–1, –3)

Example 4 Reflections in a Coordinate Plane b.The y- axis is the perpendicular bisector of AD and BE, so the reflection of AB in the y- axis is DE. A is reflected onto D. A(–4, 1) D(4, 1) B is reflected onto E. B(–1, 3) E(1, 3) Which segment is the reflection of AB in the y- axis? Which point corresponds to A ? to B ?

Your Turn: Identify Reflections ANSWER no ANSWER yes; the y- axis ANSWER yes; the x- axis Tell whether the red figure is a reflection of the blue figure. If the red figure is a reflection, name the line of reflection

Reflecting a Figure Use these same steps to reflect an entire figure on the coordinate plane: –Identify which axis is the line of reflection (the mirror). –Individually reflect each endpoint of the figure. –Connect the reflected points.

What letter would you get if you reflected each shape in its corresponding mirror line?

Now it’s your turn… On your worksheet, reflect every shape in the corresponding mirror line. Use tracing paper to help you. All the shapes should fit together to form a word. Draw in pencil in case you make any mistakes.

Symmetry

What is a line of symmetry? A line on which a figure can be folded so that both sides match. A line of symmetry is a line of reflection. A figure may have more than one line of symmetry.

Real-World Symmetry Connection Line symmetry can be found in works of art and in nature. Some figures, like this tree, Others, like this snowflake, have only one line of symmetry. have multiple lines of symmetry. Vertical symmetry Vertical, horizontal, & diagonal symmetry

Which of these flags have a line of symmetry? United States of America Canada MarylandEngland

What about these math symbols? Do they have symmetry?

Example 5 Determine Lines of Symmetry Determine the number of lines of symmetry in a square. SOLUTION Think about how many different ways you can fold a square so that the edges of the figure match up perfectly. ANSWER A square has four lines of symmetry. vertical foldhorizontal folddiagonal fold

Example 6 Determine Lines of Symmetry Determine the number of lines of symmetry in each figure. a. b. c. SOLUTION 2 lines of symmetry a. no lines of symmetry b. 6 lines of symmetry c.

How many lines of symmetry to these regular polygons have? Do you see a pattern?

Example 7 Use Lines of Symmetry Mirrors are used to create images seen through a kaleidoscope. The angle between the mirrors is  A. Find the angle measure used to create the kaleidoscope design. Use the equation m  A =, where n is the number of lines of symmetry in the pattern. n 180° a.b.c.

Example 7 Use Lines of Symmetry SOLUTION a. The design has 3 lines of symmetry. So, in the formula, n = 3. m  A = = = 60° n 180° 3 b. The design has 4 lines of symmetry. So, in the formula, n = 4. m  A = = = 45° n 180° 4 c. The design has 6 lines of symmetry. So, in the formula, n = 6. m  A = = = 30° n 180° 6

Your turn: Determine Lines of Symmetry Determine the number of lines of symmetry in the figure ANSWER 1 2 4

Assignment Pg : #1 – 39 odd, 43 – 49 odd.