Transformations Similarity and Congruence UMI: July 19, 2016

Transformation A transformation is a process which changes the position (and possibly the size and orientation) of a shape. There are four types of transformations: Rotation (Turn) Reflection (Flip) Translation (Slide) Enlargement (Resize)

Rotation (Turn) Rotation (also known as Turn) The distance from the center to any point on the shape stays the same. Every point makes a circle around the center. Changes the orientation of the shape Changes the position of the shape Everything else stays the same.

Example: Rectangle A′B′C′D′ is the image of rectangle ABCD after which of the following rotations? Find the new coordinate of the A′B′C′D′ A 90° clockwise rotation about the origin A 180° rotation about the origin A 90° counterclockwise rotation about the origin A 90° counterclockwise rotation about the point (1, 1)

Solution: Solution: C is correct. By joining A and A′ to the origin O, it is clear that this is a 90° counterclockwise rotation about the origin. The same distance from the origin.(Do the same for the other points and their images to see for yourself). The new coordinate is −1,2 , −3,2 , −3,5 , −1,5

Practice 1: The square is rotated one complete turn about the point O. Which of the following shows the new position of the square? If it's a complete turn, then the square will finish in the same position that it started, so the answer is D.

Practice 2: The square is rotated half a complete turn about the point O. Which of the following shows the new position of the square? Find the new coordinate of the square. The point O stays in the same place. But the other corners of the square go to the opposite side of O. So the answer is B. The new coordinate is 0,0 , 4,0 , −4，−4 , 0，−4 0，−4

Practice 3: When this 'L'-shape is rotated about the origin (0,0) by 90°anticlockwise (counterclockwise), which one of these would it look like?

Practice 3: The correct answer is C shows a rotation of 90° clockwise about (0,0) shows a rotation of 180° about (0,0) shows a rotation of 90° anticlockwise about (0,0) shows a rotation of 90° anticlockwise, but the center of rotation is (1,1) The correct answer is C

Reflection(Flip) Reflection (also known as Flip) in a line produces a mirror image in which corresponding points on the original shape and the mirror image are always the same distance from the mirror line. Every point is the same distance from the central line ! The reflection has the same size as the original image The central line is called the Mirror Line 

How Do I Do It Myself? Measure from the point to the mirror line (must hit the mirror line at a right angle) Measure the same distance again on the other side and place a dot. Then connect the new dots up!

Practice 1: The square is flipped across the line L. Which of the following shows the new position of the square? It's like reflecting in a mirror. So the new square must be on the opposite side of L and the same distance away from L. The correct answer is A.

Practice 2: Rectangle A′B′C′D′ is the image of rectangle ABCD after reflection in which of the following lines? The x axis The y axis The line y = x The line y = -x

Solution: The correct answer is C. Each point and its image must be the same distance from the mirror line, which is y = x

Practice 3: The rectangle is reflected in the line y = 4.Which one of the following shows the correct image? A shows a reflection in the line x = -5 B shows a reflection in the y axis, or a reflection in the x axis C shows a reflection in the line y = -3 D correctly shows a reflection in the line y = 4

Translation (Slide) Translation (also known as Slide) moves a shape by sliding it up, down, sideways or diagonally, without turning it or making it bigger or smaller. Without rotating, resizing or anything else, just moving Every point of the shape must move: the same distance in the same direction. Translation can be written down as the coordinates change 𝑥,𝑦 → 𝑥+𝑎,𝑦+𝑏

Example: Find the new image of triangle ABC using the translation: 𝑥,𝑦 → 𝑥−2,𝑦+3) Solution : 𝐴 1,1 →𝐴‘ −1,4 𝐵 6,5 →𝐵‘ 4,8 𝐶 7,1 →𝐶‘ 5,4

Practice 1: Solution : Write a rule to describe the translation: Let’s see that: 𝑌 4,1 →𝑌′ −6,−2 The rule for this translation is 𝑥,𝑦 → 𝑥−10,𝑦−3

Practice 2: The L-shape A′B′C′D′E′F′ is the image of the L-shape ABCDEF after which of the following translations? 5 units in the negative x direction and 8 units in the negative y direction 8 units in the negative x direction and 5 units in the negative y direction 8 units in the positive x direction and 5 units in the positive y direction 5 units in the positive x direction and 8 units in the positive y direction

Solution: The correct answer is D. The point A has moved 5 units in the positive x direction and 8 units in the positive y direction. Similarly, for each of the other points. This is denoted by the translation vector 5 8

Practice 3: The 'L'-shape is translated 2 units in the positive x direction and 4 units in the positive y direction. Which one of the following shows the correct image? C correctly shows a translation of 2 units in the positive x direction and 4 units in the positive y direction.

Similar Two shapes are Similar when the only difference is size (and possibly the need to move, turn or flip one around). Similar figures  have equal corresponding angles the length of their corresponding sides have equal ratios

Determine whether the polygons are similar. Justify your answer Example: Determine whether the polygons are similar. Justify your answer Solution : The corresponding angles are equal. 7 10.5 = 3 4.5 = 7 10.5 = 3 4.5 , the measures of the sides of the polygons are proportional. Therefore, the polygons are similar.

Determine whether the polygons are similar. Justify your answer Example: Determine whether the polygons are similar. Justify your answer Solution : 4 3 = 8 6 = 4 3 = 8 6 , the measures of the sides of the polygons are proportional. The corresponding angles are not equal. Therefore, the polygons are not similar.

Example: Solution : The triangles are similar. Find the value of x Write proportions using corresponding parts. Then solve to find the missing measure 𝑥 4 = 4 8 Solve 𝑥=2

Practice 1: The two trapezoids are similar. What is the value of x? Solution : The small trapezoid is a scaled-down version of the large trapezoid, but is also a mirror image. So the side x corresponds to the length 4 in the larger trapezoid. Write proportions using corresponding parts. Then solve to find the missing measure 𝑥 4 = 10 16 Solve 𝑥=2.5

Practice 2: Solution : BC is parallel to DE. What is the length of BC? Because BC is parallel to DE, pairs of corresponding angles are equal. So triangles ABC and ADE have the same angles and are similar triangles. Write proportions using corresponding parts. Then solve to find the missing measure 𝐵𝐶 9.9 = 5 9 𝐵𝐶= 5 9 ×9.9=5.5

Congruent If one shape can become another using Turns, Flips and/or Slides, then the shapes are Congruent.  Congruent figures  The same size The same shape. They are similar figures that are equal in size

Example: These figures are congruent. Find the measure of angle I. Solution : Since the figures are congruent, all angles have the same measure. According to the figures; m∠𝑄= 42 𝑜 ,𝑠𝑜 𝑚∠𝐼= 42 𝑜

Example: The two quadrilaterals are congruent. Which side in quadrilateral ABCD corresponds to WZ in quadrilateral WXYZ? Solution : WZ faces the angles marked with two arcs and four arcs. DA also faces the angles marked with two arcs and four arcs. So DA corresponds to WZ.

Thank You References: A problem solving approach to mathematics for elementary school teachers by Billstein, Libeskind, Lott Math is Fun : https://www.mathsisfun.com/ Emathematics: http://www.emathematics.net/