Create a design (image) on the graph paper on the last page, making sure at least 3 vertices land on whole number (integer) coordinates, in the upper left quadrant (Quadrant II). Place a piece of patty paper over it, with the right, lower corner of the patty paper at the origin (0, 0) and trace it. Label three of the vertices as A, B and C, on both the original design (image) on the graph paper, and on the patty paper.

2) Place your patty paper over the original design (image), with the lower right corner at the origin (0, 0). Record the original vertices of your design (image) in Reflection 1, A, B and C in the table on the following page. Flip, or reflect the patty paper over the x-axis and record the new position of coordinates A, B and C as vertices , , in the table. (These are three of the vertices of a reflection of our original image, and we say that the new vertices are A prime, B prime and C prime.) Plot and label , , and draw the remainder of your shape.

3) Complete Reflection 1 for this action recording the action taken and any observations you notice about the vertices (how did they change, or not change) and/or visually what happened to the shape.  

4) Put your design (image), or patty paper back on the originally traced one. Repeat Step 2 and Step 3, but this time flip, or reflect over the y-axis instead of the x-axis and record your work in Reflection 2.

5) Put your design (image) or patty paper back on the originally traced one in Quadrant II. Now draw the line representing y=x.

6) Place your patty paper over the original design (image), with the lower right corner at the origin (0, 0). Record the original vertices of your design (image) in Reflection 3, A, B, C. Keeping the corner of the patty paper that is at the origin (0, 0), Flip, or reflect the patty paper over the line y = x and record the new position of coordinates A, B and C as vertices , , in the table. (Your patty paper should now be in Quadrant IV.) Plot and label , , and draw the remainder of your image.

7) Complete Reflection 3 for this action recording the action taken and any observations you notice about the vertices (how did they change, or not change) or visually what happened to the shape.

WARM-UP 8 in. Perimeter = _____ 7 in. Area = _____ 12 in. 4 cm Circumference = _____ Area = _____ Perimeter = _____ 5 in Area = _____ 17 ft 25 mm 14 ft

SOLUTIONS Perimeter = _____ 30 in. 7 in. Area = _____ 56 sq. in. 4 cm SOLUTIONS Circumference = _____ 25.12 cm Area = _____ 50.24 sq. cm Perimeter = 40_ 40 mm 119 sq. ft Area = _____ 17 ft 25 mm 14 ft

Find the Perimeter and area A = ½ (b1+b2)h A = ½ bh A =L w A=Лr2 ÷ 2 A = (58) (40) A = ½(40+76 )58 A = ½(58) (36) A = (3.14) (20)2 A = 2,320 units2 A = 3364 units2 A =1,256 ÷ 2 units2 A = 1044 units2 A =628 units2 A = 3364 units2 + 628 units2 =3,992 units2

Congruent Figures Two figures have the same shape & same size) X and Y are congruent. X Y If two figures are congruent, then they will fit exactly on each other. Find out by inspection the congruent figures among the following. A B C D E F G H B, D ; C, F

TRANSFORMATIONS

Transformations Image (transformed figure) Changes position or orientation of a figure. (preserves size & shape but changes location) Each point of original figure is paired with exactly one point of its image on the plane. Image (transformed figure) (congruent to original figure.) We can identify a symmetry as a transformation of the plane that moves the pattern so that it falls back on itself. The only transformations that we'll consider are those that preserve distance, called isometries. (Self-similar fractals have symmetries on different scales, and so other transformations must be considered to understand them.) There are four kinds of planar isometries: translations, rotations, reflections, and glide reflections indicated with “prime” notation

3 Common Transformations 1. Translation, (to slide ) 2. Reflection, (flip or mirror image ) Read slide… There are other transformations … really combinations of these three… click when completed... 3. Rotation, (a turn a side around a point.)

Reflection

REFLECTION divides figure into congruent parts FLIP Line of symmetry- (splits image down middle. Creates a mirror image.)

LINE OF SYMMETRY Figure on the right shows a symmetric figure with l being the axis of symmetry. Are there are any congruent figures. The line l divides the figure into 2 congruent figures, i.e. and are congruent figures. Thus, 2 congruent figures. X

Creates a mirror image.) REFLECTION FLIP Line of symmetry- Creates a mirror image.)

Reflection (flip) specified by a line of reflection, like a mirror. Can be: vertical Horizontal diagonal A reflection fixes one line in the plane, called the axis of reflection, and exchanges points on one side of the axis with points on the other side of the axis at the same distance from the axis. In the example you see a diagonal axis in white. The double-ended red arrows are supposed to suggest the reflection. It's called a "reflection" because similar things happen with a reflection in a mirror. REFLECTION: When you look in a mirror, your image is reflected back at you. If you imagine that your image actually exists on the other side of the mirror, you get some idea of the mathematical definition of reflection or mirror image. We can reflect figures by folding along the mirror line as you see below. specified by a line of reflection, like a mirror.

“crawl before you climb” Reflect over the x axis (-10,-3) (-2,-4) (0,-1) “crawl before you climb”

“crawl before you climb” Reflect over the y axis (10,3) (2,4) (0,1) “crawl before you climb”

“crawl before you climb” Reflect over the y axis (-10,-3) (-2,-4) (0,-1) “crawl before you climb”

Symmetry of the Alphabet Asymmetric =(not mirror image of graph along axis) Symmetric = (mirror image of graph along axis) Symmetry of the Alphabet Sort capital letters of alphabet into groups according to symmetries Divide letters into two categories: symmetrical not symmetrical Symmetrical: A, B, C, D, E, H, I, K, M, N, O, S, T, U, V, W, X, Y, Z Not Symmetrical: F, G, J, L, P, Q, R

SYMMETRY

2 types of symmetry: 1. Reflectional Symmetry 2. Rotational Symmetry

Symmetry Reflection Rotational Symmetry (rotates around a set point) Look at the figure at right and does it have: Symmetry? Rotational symmetry? If so what angles? Rotate the figure around the center Reflection Rotational Symmetry (rotates around a set point)

A figure is turned about a point and it coincides with the original, it has rotational symmetry. How many degrees has this figure rotated? 120 degrees How? 360/3 = 120

point symmetry (figure with rotational symmetry of 180o)

A rotation of 360 degrees is called a rotational identity.

Every figure has symmetry of rotational identity. Therefore, every figure must have at least one symmetry.

Determine if each figure has rotational symmetry. If so, list the degree of turn needed to complete the rotation. Point Symmetry 90 degrees Rotational Identity 60 degrees 90 degrees

Reflectional Symmetry is also called Line Symmetry. Horizontal Vertical Vertical and Horizontal

GLIDE REFLECTION transformation (movement of one figure where each point of original figure is paired with exactly one point of its image on the plane. Isometry (transformation that preserves size & shape but changes location) Image (always congruent to original figure.)

Glide Reflection Involves more than one translation Combines reflection and translation Required: Reflection line must be parallel to direction of slide.

Glide reflection glide reflection (special product of a reflection & a translation along line of reflection.) only type of symmetry involving more than one step. The fourth kind of isometry, the glide reflection, is not nearly as easy to see as the other three. It's composed of a reflection across an axis and a translation along the axis. GLIDE REFLECTION: This transformation combines translations and reflections. A glide reflection occurs when you slide an image in one direction and then reflect it over a line.

What type is each transformation? All points slide same distance in same direction. Translation picture is “turned”. Rotation image is “flipped” across a line. Reflection 2-step process: slide & flip. Glide reflection

Group Activity Choose a letter (other than R) with no symmetries On a piece of paper perform the following tasks on the chosen letter: rotation translation reflection glide reflection

Transformations Four Types: TO Slide, flip, or turn a figure/pattern. Reflection Rotation Translation Glide Reflection Isometry (transformation that preserves size & shape but changes location)

Transformation and Congruence B) 11.1 The Meaning of Congruence 1B_Ch11(54) Example Transformation and Congruence B) ‧ When a figure is translated, rotated or reflected, the image produced is congruent to the original figure. When a figure is enlarged or reduced, the image produced will NOT be congruent to the original one. Index 11.1 Index

(c) Enlargement No (e) ____________ Reduction No

SYMMETRY Activity Take out a piece of paper. Pour 5 drops of paint in the center of paper. Fold the paper in half. Open the paper and let it dry.

Reflection (flip or mirror reflection) A reflection is determined by a line in the plane called the line of reflection. Each point P of the plane is transformed to the point P’ on the opposite side of the line of reflection and the same distance from the line of reflection.

1. The image of point (3,-5) under the translation that shifts (x,y) to (x-1,y-3) is..... A. (-4,8) B. (2,8) C. (-3,15) D. (2,-8) image of point (3,-5) under translation that shifts (x,y) to (x-1,y-3) is (2,-8) 2. A translation maps (x,y) (x+1,y+2) what are the coordinates of B (-2,4) after translation? coordinates of B(-2,4) after translation: (-1,6). 3. What is the image of point P(-3,2) under the transformation T(2,6)? T(-2,6) means add -2 to the x-value (-3) & +6 to the y-value (2). image of point P is (-5,8).

What is the image of point P(4,2) under the transformation T(-2,2)? The image of point P(4,2) under the transformation T(-2,2) is (2,4). What is the image of point P(-2,-7) under the transformation T(6,4)? The image of point P(-2,-7) under the transformation T(6,4) is (4,-3). If the point (4,1) has a translation of (-2,4), what are the coordinates of pt. (-1,5) under the same translation? To solve this problem, you have to add (-2,4) to the point (-1,5). Always add x-value with x-value and vice-versa (For example: add -2 to -1 and add 4 to 5.) The answer is (-3,9).

If the coordinates of the vertices of triangle ABC are A(-4,-1), B(-1,5) and C(2,1), what are the coordinates of triangle A'B'C', the translation of triangle ABC, under T(4,3)? The coordinates of triangle A‘ B‘ C' are A'(0,2), B'(3,8) and C'(6,4). How do we translate points & figures in a coordinate plane?