Transformation in Geometry

Transformation A transformation changes the position or size of a polygon on a coordinate plane

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

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

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

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.

Translation 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: C A B object C A B C A B C A B C A B C A B C A B C A B C’ A’ B’ image Every point in the shape moves the same distance in the same direction. object

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

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

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

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 negative numbers are used for movements down or to the left.

Translation golf

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

In a reflection, a mirror image of the polygon is formed across a line called a line of symmetry. No change in size. The orientation of the shape changes. Corresponding points are an equal distance from the line of symmetry A reflection across the x -axis --change the sign of the y coordinate. A reflection across the y-axis – change the sign of the x-coordinate.

Reflection over X axis – multiply the y coordinate by – 1 (x, y) ( x, –y ) (3, –4) ( 3, 4 ) (x, y) (y, x) (x, y ) ( –x, y) (3, –4) ( –3, –4) (3, –4) ( –4, 3) (3, –4) (–3, 4 ) (x, y) (–x, –y ) Origin – multiply both coordinates by -1 y = x – switch the x and y coordinatesY axis – multiply the x coordinate by – 1

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.

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

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

Reflecting shapes

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.

Reflect this shape

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.

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

REFLECTIONAL SYMMETRY The line created by the fold is the line of symmetry. A shape can have more than one line of symmetry. Where is the line of symmetry for this shape? How can I fold this shape so that it matches exactly? Line of Symmetry

REFLECTIONAL SYMMETRY How many lines of symmetry does each shape have? Do you see a pattern?

REFLECTIONAL SYMMETRY Which of these flags have reflectional symmetry? United States of America Mexico Canada England

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

– 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)

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

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

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

ROTATION If a shape spins 360 , how far does it spin? 360 

ROTATION If a shape spins 180 , how far does it spin? 180  Rotating a shape 180  turns a shape upside down.

ROTATION If a shape spins 90 , how far does it spin? 90 

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

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

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

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? 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. 90 

ROTATIONAL SYMMETRY 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  A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.

ROTATIONAL SYMMETRY 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. A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape.

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

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.

Finding the order of rotational symmetry

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

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 counterclockwise rotation. 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°.

Dilation A transformation where a figure changes size.

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.

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.

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.

Enlargement 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 B C A’ B’ C’ = B’C’ BC = A’C’ AC = the scale factor A’B’ AB 4 cm 6 cm 8 cm 9 cm 6 cm 12 cm 6 4 = 12 8 = 9 6 = 1.5

Congruence and similarity Is the image of an object that has been enlarged congruent to the object? Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal. In an enlarged shape the corresponding angles are the same but the lengths are different. The image of an object that has been enlarged is not congruent to the object, but it is similar. In maths, two shapes are called similar if their corresponding angles are equal. Corresponding sides are different lengths, but the ratio in lengths is the same for all the sides.

Find the scale factor What is the scale factor for the following enlargements? B B’ Scale factor 3

Find the scale factor What is the scale factor for the following enlargements? Scale factor 2 C C’

Find the scale factor What is the scale factor for the following enlargements? Scale factor 3.5 D D’

Find the scale factor What is the scale factor for the following enlargements? Scale factor 0.5 E E’

Using a center of enlargement To define an enlargement we must be given a scale factor and a center of enlargement. For example, enlarge triangle ABC by scale factor 2 from the centre of enlargement O: O A C B OA’ OA = OB’ OB = OC’ OC = 2 A’ C’ B’

Using a centre of enlargement Enlarge parallelogram ABCD by a scale factor of 3 from the centre of enlargement O. O D A B C OA’ OA = OB’ OB = OC’ OC = 3= OD’ OE A’D’ B’C’

Exploring enlargement

Enlargement on a coordinate grid The vertices of a triangle lie on the points A(2, 4), B(3, 1) and C(4, 3). The triangle is enlarged by a scale factor of 2 with a centre of enlargement at the origin (0, 0) A(2, 4) B(3, 1) C’(8, 6) A’(4, 8) B’(6, 2) What do you notice about each point and its image? y x C(4, 3)

Coordinate Dilation Each point is multiplied by k to find the dilation.

Enlargement on a coordinate grid The vertices of a triangle lie on the points A(2, 3), B(2, 1) and C(3, 3). The triangle is enlarged by a scale factor of 3 with a centre of enlargement at the origin (0, 0). What do you notice about each point and its image? A(6, 9)C’(9, 9) B’(6, 3) A(2, 3) B(2, 1) C(3, 3)

Combining reflections An object may be reflected many times. In a kaleidoscope mirrors are placed at 60° angles. Shapes in one section are reflected in the mirrors to make a pattern. How many lines of symmetry does the resulting pattern have? Does the pattern have rotational symmetry?

Parallel mirror lines What happens when an object is reflected in parallel mirror lines placed at equal distances?

Parallel mirror lines Reflecting an object in two parallel mirror lines is equivalent to a single translation. M1M1 M2M2 A A’A’’ Suppose we have two parallel mirror lines M 1 and M 2. We can reflect shape A in mirror line M 1 to produce the image A’. We can then reflect shape A’ in mirror line M 2 to produce the image A’’. How can we map A onto A’’ in a single transformation?

Perpendicular mirror lines M2M2 M1M1 A A’ A’’ We can reflect shape A in mirror line M 1 to produce the image A’. We can then reflect shape A’ in mirror line M 2 to produce the image A’’. How can we map A onto A’’ in a single transformation? Reflection in two perpendicular lines is equivalent to a single rotation of 180°. Suppose we have two perpendicular mirror lines M1 and M2.

Combining rotations Suppose shape A is rotated through 100° clockwise about point O to produce the image A’. O A A’ 100° Suppose we then rotate shape A’ through 170° clockwise about the point O to produce the image A’’. How can we map A onto A’’ in a single transformation? 170° A’’ To map A onto A’’ we can either rotate it 270° clockwise. Two rotations about the same centre are equivalent to a single rotation about the same centre.

Combining translations Suppose shape A is translated 4 units left and 3 units up. Two or more translations are equivalent to a single translation. A A’’ Suppose we then translate A’ 1 unit to the left and 5 units down to give A’’. A’ How can we map A to A’’ in a single transformation? We can map A onto A’’ by translating it 5 units left and 2 units down.

Transformation shape sorter

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