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Transformation in Geometry

Transformation in Geometry

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Transformation in Geometry

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  1. Transformation in Geometry

  2. Transformation A transformation changes the position or size of a shape on a coordinate plane

  3. TRANSFORMATIONS CHANGE THE POSTION OF A SHAPE CHANGE THE SIZE OF A SHAPE TRANSLATION ROTATION REFLECTION DILATION Change in location Turn around a point Flip over a line Change size of a shape

  4. Renaming Transformations It is common practice to name transformed polygon using the same letters with a “prime” symbol: It is common practice to name a polygon using capital letters for each of its vertices: The original polygon is often called the Pre-image The transformed polygon is often called the image

  5. Translation A transformation that moves each point in a figure the same distance in the same direction.

  6. In a translation a figure slides up or down, or left or right. No change in shape, size or the direction it is facing. • The location is the only thing that changes. They are sometimes called “slides” • In graphing translation, all x and y coordinates of a translated figure change by adding or subtracting.

  7. Translation A’ image A A A A A A A A C’ B’ object object object object object object object object C C C C C C C C B B B B B B B B When an object is moved in a straight line in a given direction we say that it has been translated. For example, we can translate triangle ABC 5 squares to the right and 2 squares up: object Every point in the shape moves the same distance in the same direction.

  8. Describing translations When we describe a translation we always give the movement left or right first followed by the movement up or down. We can describe translations using function notation. For example, describes a translation of triangle ABC as 3 right and 4 down. As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left.

  9. Translations on a coordinate grid A(5, 7) C(–2, 6) B(3, 2) A’(2, –1) C’(–5, –2) B’(0, –6) y The vertices of a triangle lie on the points A(5, 7), B(3, 2) and C(–2, 6). 7 6 5 4 3 2 Translate the shape 3 squares left and 8 squares down. Label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 What do you notice about each point and its image? –3 –4 –5 –6 –7

  10. Translations on a coordinate grid y The coordinates of vertex A of this shape are (–4, –2). 7 6 5 4 3 When the shape is translated the coordinates of vertex A’ are (3, 2). 2 A’(3, 2) 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 What translation will map the shape onto its image? A(–4, –2) –3 –4 –5 –6 –7 7 right 4 up

  11. Translations on a coordinate grid y 7 6 5 4 3 2 1 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x –1 –2 –3 –4 –5 –6 –7 The coordinates of vertex A of this shape are (3, –4). When the shape is translated the coordinates of vertex A’ are(–3, 3). A’(–3, 3) What translation will map the shape onto its image? A(3, –4) 6 left 7 up

  12. Translation golf

  13. Reflection A transformation where a figure is flipped across a line such as the x-axis or the y-axis.

  14. In a reflection, a mirror image of the figure is formed across a line called a line of symmetry. • No change in size. The orientation of the shape changes. • In graphing, a reflection across the x -axis changes the sign of the y coordinate. A reflection across the y-axis changes the sign of the x-coordinate.

  15. Reflection An object can be reflected in a mirror line or axis of reflection to produce an image of the object. For example, Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.

  16. Reflecting shapes A B C D If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’. A’ B’ object image C’ D’ mirror line or axis of reflection The image is congruent to the original shape.

  17. Reflecting shapes A A’ B B’ object image C C’ D D’ mirror line or axis of reflection If we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line.

  18. Reflecting shapes

  19. Reflecting shapes using tracing paper Suppose we want to reflect this shape in the given mirror line. Use a piece of tracing paper to carefully trace over the shape and the mirror line with a soft pencil. When you turn the tracing paper over you will see the following: Place the tracing paper over the original image making sure the symmetry lines coincide. Draw around the outline on the back of the tracing paper to trace the image onto the original piece of paper.

  20. Reflect this shape

  21. 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.

  22. REFLECTIONAL SYMMETRY Line of Symmetry 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. The two halves are exactly the same… They are symmetrical. The two halves make a whole heart.

  23. REFLECTIONAL SYMMETRY The line created by the fold is the line of symmetry. How can I fold this shape so that it matches exactly? A shape can have more than one line of symmetry. Where is the line of symmetry for this shape? I CAN THIS WAY NOT THIS WAY Line of Symmetry

  24. REFLECTIONAL SYMMETRY How many lines of symmetry does each shape have? 3 4 5 Do you see a pattern?

  25. Canada England REFLECTIONAL SYMMETRY Which of these flags have reflectional symmetry? No United States of America No Mexico

  26. Rotation A transformation where a figure turns about a fixed point without changing its size and shape.

  27. In a rotation, figure turns around a fixed point, such as the origin. • No change in shape, but the orientation and location change. • Rules for 90 degrees rotation about the origin- Switch the coordination of each point. Then change the sign of the y coordinate. • Ex. A (2,1) to A’ ( 1,-2)

  28. Describing a rotation The angle of rotation. The direction of rotation. The centerof rotation. A rotation occurs when an object is turned around a fixed point. To describe a rotation we need to know three things: For example, ½ turn = 180° ¼ turn = 90° ¾ turn = 270° For example, clockwise or counterclockwise. This is the fixed point about which an object moves.

  29. ROTATION What does a rotation look like? center of rotation A ROTATION MEANS TO TURN A FIGURE

  30. ROTATION The triangle was rotated around the point. This is another way rotation looks center of rotation A ROTATION MEANS TO TURN A FIGURE

  31. ROTATION If a shape spins 360, how far does it spin? 360 All the way around This is called one full turn.

  32. ROTATION If a shape spins 180, how far does it spin? Rotating a shape 180 turns a shape upside down. Half of the way around 180 This is called a ½ turn.

  33. ROTATION If a shape spins 90, how far does it spin? One-quarter of the way around 90 This is called a ¼ turn.

  34. ROTATION Describe how the triangle A was transformed to make triangle B A B Triangle A was rotated right 90 Describe the translation.

  35. ROTATION Describe how the arrow A was transformed to make arrow B B A Arrow A was rotated right 180 Describe the translation.

  36. ROTATION When some shapes are rotated they create a special situation called rotational symmetry. to spin a shape the exact same

  37. 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? 90 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.

  38. 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 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

  39. 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 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.

  40. ROTATION SYMMETRY Does this shape have rotational symmetry? Yes, when the shape is rotated 120 it is the same. Since 120  is less than 360, this shape HAS rotational symmetry 120

  41. Rotational symmetry An object has rotational symmetry if it fits exactly onto itself when it is turned about a point at its centre. The order of rotational symmetry is the number of times the object fits onto itself during a 360° turn. If the order of rotational symmetry is one, then the object has to be rotated through 360° before it fits onto itself again. Only objects that have rotational symmetry of two or more are said to have rotational symmetry.

  42. Finding the order of rotational symmetry

  43. Rotational symmetry What is the order of rotational symmetry for the following designs? Rotational symmetry order 4 Rotational symmetry order 3 Rotational symmetry order 5

  44. Determining the direction of a rotation Sometimes the direction of the rotation is not given. If this is the case then we use the following rules: A positive rotation is an counterclockwiserotation. A negative rotation is an clockwise rotation. For example, A rotation of 60° = an anticlockwise rotation of 60° A rotation of –90° = an clockwise rotation of 90° Explain why a rotation of 120° is equivalent to a rotation of –240°.

  45. Dilation A transformation where a figure changes size.

  46. Dilation • In dilation, a figure is enlarged or reduced proportionally. No change in shape, but unlike other transformation, the size changes. • In graphing, for dilation, all coordinates are divided or multiplied by the same number to find the coordinates of the image.

  47. Dilation Dilation is altering the size of the figure by k = n . The dilation of this figure is k = 3. The image is 3 times bigger than the preimage. DILATION – A transformation that alters the size of a figure, but not its shape.

  48. Enlargement A’ A Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. We say that shape A has been enlarged by scale factor 2.

  49. Enlargement A’C’ B’C’ A’B’ 6 12 9 AB BC AC 4 8 6 When a shape is enlarged the ratios of any of the lengths in the image to the corresponding lengths in the original shape (the object) are equal to the scale factor. A’ A 6 cm 4 cm 9 cm 6 cm B B’ 8 cm C 12 cm C’ = = = the scale factor = = = 1.5