Starter 4 2 1 0.45 5000 Convert the following: 4000 m = __________km 20 mm = __________cm 100 cm = __________ m 45 cm = __________ m 5 km = __________ m 4 2 1 0.45 5000 ÷ 10 ÷ 1000 ÷ 100 cm m km mm × 10 × 100 × 1000

Geometry Transformations

Why is it important that an airplane is symmetrical? Are the freight containers mirror images of each other? How are the blades of the engine symmetrical? Reflection Rotation

Which aircraft does not have a symmetrical seating plan? Is it possible for a symmetrical aircraft to have an odd number of seats in a row? Which aircraft has a 2-3-2 seating plan in economy class?

Symmetry

Axes of Symmetry A line of symmetry divides a shape into two parts, where each part is a mirror image of the other half. Example: 2 axes of symmetry

Line Symmetry - How many axes of Symmetry can you find?

Note 1: Order of Rotational Symmetry The order of rotational symmetry is how many times the object can be rotated to ‘map’ itself.  (through an angle of 360° or less)

Rotational order of Symmetry 6 2 4 8 3 5

Which one of these cards has a Rotational Order of Symmetry = 2?

Note 1: Total order of symmetry (Line Symmetry) Total order of Symmetry   Number of Axes of Symmetry Order of Rotational Symmetry = +

Shape Axes of Symmetry Order of Rotational Symmetry Total Order of Symmetry 4 8 4 2 2 1 1 2 6 6 12

Task ! Choose 3 objects in the room and describe their axes of symmetry, order of rotational symmetry and total order of symmetry. Can you find an object with a total order of symmetry greater than 4 ?

Note 1: Total order of symmetry Total order of symmetry = the number of axes of symmetry + order of rotational symmetry.  The number of axes of symmetry is the number of mirror lines that can be drawn on an object. The order of rotational symmetry is how many times the object can be rotated to ‘map’ itself.   (through an angle of 360° or less)

Note 2: Reflection A point and its image are always the same distance from the mirror line If a point is on the mirror line, it stays there in the reflection. This is called an invariant point.

Reflect across the y-axis B B’ C C’ E E’ D D’

Reflect A’B’C’D’E’ across the line y=x Which are the invariant points? B’ D’’ and E’’ C’ E’ A’’ E’’ D’ D’’ C’’ B’’

Reflection To draw an image: Measure the perpendicular distance from each point to the mirror line. Measure the same perpendicular distance in the opposite direction from the mirror line to find the image point. (often it is easier to count squares).  e.g. Draw the image of PQR in the mirror line LM. IGCSE Ex 3 pg 279-280 Ex 4 pg 280-281

Analyze the ALPHABET ALPHABET Notice the letter B, H and E are unchanged if we take their horizontal mirror image? Can you think of any other letters in the alphabet that are unchanged in their reflection? What is the longest word you can spell that is unchanged when placed on a mirror? Can you draw an accurate reflection of your own name?

To draw a mirror line between a point and its reflection: 1. Construct the perpendicular bisector between the point and its image.   e.g. Find the mirror line by which B` has been reflected from B.

Practice Drawing a Reflections and mirror lines! Count squares or measure with a ruler Handouts – Reflection, Mirror lines Homework - Finish these handouts.

Rotation

What are these equivalent angles of rotation? Rotations are always specified in the anti-clockwise direction. What are these equivalent angles of rotation? 270° Anti-clockwise is _______ clockwise 180 ° Anti-clockwise is ______ clockwise 340 ° Anti-clockwise is ______ clockwise

Drawing Rotations ¼ turn clockwise = 90º clockwise B C Rotate about point A ¼ turn clockwise = 90º clockwise A B’ D D’ C’

To draw images of rotation: Measure the distance from the centre of rotation to a point. Place the protractor on the shape with the cross-hairs on the centre of rotation and the 0o towards the point. Mark the wanted angle, ensuring to mark it in the anti-clockwise direction. Measure the same distance from the centre of rotation in the new direction. Repeat for as many points as necessary.

Examples Rotate flag FG, 180 about O Draw the image A`B`C`D` of rectangle ABCD if it is rotated 90o about point A. A

Rotation In rotation every point rotates through a certain angle about a fixed point called the centre of rotation. Rotation is always done in an anti-clockwise direction. A point and its image are always the same distance from the centre of rotation. The centre of rotation is the only invariant point.

180º By what angle is this flag rotated about point C ? Remember: Rotation is always measured in the anti clockwise direction!

By what angle is this flag rotated about point C ? 90º

270º By what angle is this flag rotated about point C ? IGCSE Ex 5 pg 282-283 Ex 6 pg 283-284

Define these terms Mirror line Centre of rotation Invariant The line equidistant from an object and its image The point an object is rotated about Doesn’t change Mirror line Centre of rotation Invariant What is invariant in Reflection rotation The mirror line The size of angles and sides The area of the shape Centre of rotation Size of angles and sides

Translations Each point moves the same distance in the same direction There are no invariant points in a translation (every point moves)

( ) ( ) Vectors Vectors describe movement x y ← movement in the x direction (left and right) y ← movement in the y direction (up and down) Each vertex of shape EFGH moves along the vector ( ) -3 -6 To become the translated shape E’F’G’H’

Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`. ( ) - 4 - 2 IGCSE Ex 7 pg 285-280

Enlargement In enlargement, all lengths and distances from a point called the centre of enlargement are multiplied by a scale factor (k).

To draw an enlargement Measure the distance from the centre of enlargement to a point. Multiply the point by the scale factor and mark the point’s image point. Continue for as many points as necessary.  

Enlarge the ABC by a scale factor of 2 using the point O as the centre of enlargement.

To find the centre of enlargement Join each of the points to it’s image point. The point where all lines intersect is the centre of enlargement.

Calculating the scale factor To calculate the scale factor (k) we use the formula : Scale factor (k) = = IGCSE Ex 8 pg 287

Negative Scale Factors When the scale factor is negative, the image is on the opposite side of the centre of enlargement from the object. To draw images of negative scale factors: Measure the distance from the centre of rotation to a point. Multiply the distance by the scale factor. Measure the distance on the opposite side of the centre of rotation from the point. Repeat for as many points as necessary.

Enlarge XYZ by a scale factor of –2 about O.

Do Now: Match up the terms with the correct definition Write them into your vocab list Cuts a line into two equal parts (cuts it in half) – also called the mediator The transformed object A transformation which maps objects across a mirror line A line which intersects a line at right angles The line in which an object is reflected Image Mirror line Perpendicular Bisector Reflection

Do Now: What transformations are in each example? 1. 2. 3. Rotation Enlargement Reflection 4. 5. Translation Reflection

Combined Transformations Let A be a reflection in line x = 0 Let B be a translation of Let C be a rotation of 180° about O (0,0) Notice that, like composite functions, we work from right to left A(▲1) means ‘perform the transformation A on triangle ▲1 BA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1 CBA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1, then perform the transformation C

A(▲1) means ‘perform the transformation A on triangle ▲1 ▲2 Let A be a reflection in line x = 0

BA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1 (▲2) ▲2 ▲1 ▲3 Let B be a translation of

CBA(▲1) means ‘perform the transformation A on triangle ▲1, then perform the transformation B on the image of ▲1, then perform the transformation C ▲1 ▲2 ▲4 ▲3 Let C be a rotation of 180° about O (0,0)

Repeated Transformations CC(▲1) Let C be a rotation of 180° about O (0,0)

Inverse Transformations The inverse of a translation T with vector Is If a rotation R is 90° clockwise, then the inverse R-1 is 90° Anti-clockwise The inverse of a transformations takes the image back to the object.

Reflect the shape in the red mirror line, translate the image by the vector Enlarge the image scale factor 3, centre P Rotate 45o , centre A’’’ A IGCSE Ex 10 pg 290 P

Starter 1.) Reflection is a transformation which maps an object across a __________. 2.) In rotation, the only invariant point is called the __________________. 3.) A _______ describes the movement up and down, and across, in a translation. 4.) All of the _________ points in reflection lie on the mirror line. 5.) The area of the object, the size of the angles and the length of the sides are invariant in both rotation and ___________ mirror line centre of rotation vector invariant reflection invariant, centre of rotation, reflection, vector, mirror line

Describing Transformations using Base Vectors We can describe transformations in a matrix by analysing its effect on base vectors. The base vectors in 2D are and y I J J (0,1) 0 1 -1 0 e.g. In a transformation matrix, The 1st column is I’ (image of I) The 2nd column is J’ (image of J) (1,0) x I J’ 90° Clockwise Rotation I’ * Useful when the origin remains fixed

You try! Use base vectors to describe the transformation represented by: a.) b.) -1 0 0 1 2 0 0 2 y IGCSE Ex 14 pg 298 y J (0,1) J (0,1) (1,0) (1,0) x x I I Reflection across x=0 Enlargement – scale factor 2, C.E(0,0)

Shear ½ x x A B A B What is invariant? Area Line AB

Stretch To describe a stretch we need to know: a.) The invariant line b.) The direction of the stretch c.) The ratio of corresponding lengths y IGCSE Ex 15 pg 299 1 0 0 k The invariant line is y = 0 x The ratio of corresponding lengths is k

Koru Design Using the templates provided, or your own, create a pattern of at least 5 transformations, which consists of at least: One reflection One translation One rotation

How to Write Instructions Reflect ABCD through the mirror line (M) Translate the image A’B’C’D’ 4 cm to the right Rotate the image A’’B’’C’’D’’, 90o counter clockwise about the point P. 4 cm M P

Now its your turn! Write the appropriate instruction for each transformation, in the order that it appears. A’’’ A’’’’ A’’ P 6 cm 5 cm A A’ M

Writing Instructions ( ) → ( ) 1.) Label your object 2.) Reflect image about mirror line M 3.) Translate the image a’b’c’d’ by the vector ( ) → ( ) 4.) Rotate the image a’’b’’c’’d’’ about point P 90º M b c c’ b’ x 4 2 1 y -7 a d d’ a’ c’’ b’’ 3 b’’’ a’’’ d’’ a’’ 4 c’’’ d’’’ P

Writing Instructions ( ) to give image a’’’b’’’c’’’ 1.) Construct equilateral triangle abc (Label your object) 2.) Rotate object abc 90º about point P to give a’b’c’. 3.) Reflect image a’b’c’ through mirror line M to give image a’’b’’c’’ 4.) Translate the object a’’b’’c’’ by the vector ( ) to give image a’’’b’’’c’’’ c’’’ 4 b’’’ b a’’’ 1 a c P c’ c’’ b’ b’’ 2 3 a’ a’’ 8 M 4

Koru Design Using the templates provided, or your own, create a pattern of at least 5 transformations, which consists of at least: One reflection One translation One rotation * Write a set of instructions so another student could reproduce your pattern.