1.) Create a design (image) on the graph paper with 3 vertices land on whole number (integer) coordinates, in the upper left quadrant (Quadrant II). Label the origin (0,0) on the coordinate plane on the worksheet. 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. (Note: Draw a point at the bottom right corner of the patty (at origin) and label this point R. This will be the point about which you rotate the design.)

We would say that the image is being “rotated about Point R.” 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 Rotation 1, A, B and C in the table on the following page. Rotate the patty paper from point R, by putting your pencil at point R and turning the patty paper 90 clockwise. Record the new position of coordinates A, B and C as vertices A’ , B’, C’ in the table. (Point R should remain the same, and the patty paper should now be in Quadrant I.) Plot and label A’ ,B’ , and C’ draw the remainder of your shape. We would say that the image is being “rotated about Point R.”

3) Complete Rotation 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 patty paper design back on triangle A,B,C (in quadrant II). This time rotate your image 90 to the left (counterclockwise), by holding your pencil at point R. Record your work in Rotation 2. (Rotating the 90 left is the same as a 270 right!)

5) 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 Rotation 3, A, B, C. Keeping the corner of the patty paper that is at the origin (0, 0) and holding your pencil on point R, rotate the patty paper 180 (90 x 2, so patty paper is now in Quadrant IV). Record the new position of coordinates A, B and C as vertices A”’, B”’ and C”’ in the table. Plot and label A”’, B”’ and C”’ and draw the remainder of your image.

Complete Rotation 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.

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.)

Rotation

Rotation (turn) A rotation is a turn through an angle. specified by: center of rotation an angle measure. rotational symmetry a turn of n degrees, around a center point, A rotation fixes one point in the plane and turns the rest of it some angle around that point. In general a rotation could be by any angle, but for patterns like we have, the angle has to divide 360°, and a little more analysis finds further restrictions. In fact, the angle of rotation can only be 180°, 120°, 90°, or 60°. The order of a rotation is the number of times it has to be performed to bring the plane back to where it started. So a 60°-rotation has order 6, a 90°-rotation has order 4, a 120°-rotation has order 3, and an 180°-rotation has order 2. A 180° rotation is also called a half turn. ROTATION: To rotate is to turn about a point. When you make a left-hand turn at a corner, we say you are rotating 90 degrees about the corner.

rotates a figure around a point. Rotation To rotate must have: a center an angle rotates a figure around a point.

Rotation is simply turning about a fixed point. Here the fixed point is the origin Rotate 90 counterclockwise about the origin Rotate 180 about the origin Rotate 90clockwise about the origin

Rotate 90 degrees clockwise. (like a right turn) Hands in the air on the wheel. Left hand: x Right hand: y Make a clockwise turn. Which hand is at 12 o’clock 1st? X Rotate 90 degrees clockwise. 1. Change the sign of x 2. Switch order of x & y.

Rotate 90 degrees clockwise.

Rotate 90° clockwise

Rotate 90° clockwise

Rotate 90 degrees counterclockwise. is a left turn. Hands in the air on the wheel. Left hand: x Right hand: y Make a counterclockwise turn. Which hand is at 12 o’clock 1st? Y Rotate 90 degrees counterclockwise. 1. Change sign of y 2. Switch the order of x and y

Example: Rotate 90 degrees counterclockwise.

Rotate 90° counterclockwise

Rotate 90° counterclockwise

Rotating 180 degrees changes the sign of the x and the sign of the y. Rotate 180 degrees. Rotating 180 degrees changes the sign of the x and the sign of the y. Keep the order & change the sign of both x & y.

Example: Rotate 180 degrees.

Rotate 180°

Rotate 180°

Homework Does the Brain Good.

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

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(39) 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?

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