Transformations, Constructions and 3D Drawings Geometric Techniques Transformations, Constructions and 3D Drawings

Symmetry 1. Line Symmetry - A shape has line symmetry if it can fold directly onto itself. - The line of folding (mirror line) is called an axis of symmetry e.g. Draw in the lines of symmetry for the following shapes An infinite number of lines of symmetry 4 lines of symmetry 0 lines of symmetry 2. Rotational Symmetry - the order of rotational symmetry is the number of times a figure maps exactly onto itself during a turn of 360° - Every shape has an order of rotational symmetry of at least one. e.g. State the order of rotation for the following shapes An infinite order of rotational symmetry Order of 4 Order of 1

Point Symmetry - Occurs when an object maps onto itself after a half turn or a turn of 180° e.g. Which of the following shapes has point symmetry? YES NO 3. Total Order of Symmetry - Is the number of lines (axes) of symmetry plus the order of rotational symmetry. e.g. State the total order of symmetry for the following shapes 4 lines of symmetry 0 lines of symmetry ∞ lines of symmetry Order of 4 Order of 1 Order is ∞ Total Order of Symmetry = 8 Total Order of Symmetry is infinite Total Order of Symmetry = 1

Reflection - In reflection, an object and its image are on the opposite sides of the mirror line - The object and its image are ALWAYS the same distance from the mirror line e.g. Reflect the following objects in mirror line m. B B’ B B’ A A’ A A’ C C’ C C’ m m Properties: - Reflected lines, angles and areas remain the same size Note: We use a dash after the letters to show which is the image - Points on the mirror line are invariant - Sense or orientation is reversed

- There can also be more than one mirror line e.g. Reflect the following object in mirror line m and n. n Note: When fully describing a reflection, a mirror line must be added and discussed in the answer. m Finding the mirror line. - The mirror line is ALWAYS half way between each point and its image e.g. Draw in the mirror line for the following reflections. A B B A C C’ D D’ B’ A’ A’ B’

Rotation - To perform rotations a centre of rotation and an angle of rotation (or amount of turn) are needed - If not stated, the direction of a rotation is always anti-clockwise e.g. Rotate the object 90° clockwise about X e.g. Rotate the object 270° about X B C B A’ A D A’ B’ C’ B’ X D’ C’ A C X Properties: Note: It can be a good idea using tracing paper and turning the paper about the centre of rotation - Rotated lines, angles and areas remain the same size - Each point and its image are the same distance from the centre of rotation - The centre of rotation is invariant

Finding the centre of rotation To find the centre either: -Use tracing paper and guess location or -Find the intersection of the perpendicular bisectors of the lines joining a point to its image e.g. Find the centre of rotation A’ X A Note: When fully describing a rotation, a centre, angle and direction must be added and discussed in the answers

Enlargement - To perform an enlargement, a centre and scale factor are needed 1. Scale Factor (s.f.) - To calculate the scale factor, use the following formula Scale factor (s.f.) = length of image length of object e.g. Calculate the scale factor of the following enlargement A’ Scale factor (s.f.) = 6 3 A 9 = 2 B’ C’ B C 6 3 2. Centre of Enlargement - The centre of enlargement is at the intersection of the lines joining points to their images X A’ A e.g. Find the centre and scale factor of the following enlargement Scale factor (s.f.) = 1 2 Centre = X

3. Drawing Enlargements a) Using only the scale factor - Multiply lengths of the object by the s.f. and draw the image anywhere e.g. Enlarge the object by scale factor of 2 Remember when enlarging by only the scale factor, the image can be placed anywhere Note: When fully describing an enlargement, a centre and scale factor must be added and discussed in the instructions A A’ a) Using a centre and scale factor - Choose a point on the object and measure the distance between it and the centre - Multiply the distance by the scale factor and use it to plot a point of the image - Draw in the image making sure it is enlarged by the scale factor. e.g. Enlarge the object by scale factor of 2 and centre X Properties: - Shape, sense and angle size remain the same A’ - Lengths and areas change A - The centre of enlargement is invariant X

- An object can also shrink with an enlargement! - If the scale factor is -1 < x < 1 then the image will shrink. - If the scale factor is negative then the image ends up on the other side of the centre and reversed. e.g. Enlarge the object by scale factor of 0.5 and centre X A A’ X e.g. Enlarge the object by scale factor of -0.5 and centre X X A’ A

Translation - Is the movement of an object, without twisting, in a particular direction - The movement can be described by a vector or a direction - In a vector the top number describes sideways movement (negative = left) - The bottom number describes up/down movement (negative = down) e.g. Translate the object ‘A’ by the vector A Properties: - Translated lines, sense, angles and areas remain the same size A’ - There are no invariant points e.g. By what vector has the following object been translated? Vector = A’ Note: When fully describing a translation, a vector or direction must be added and discussed in the answer A

Constructing Perpendicular Bisectors - Perpendicular Bisectors are lines that cut each other in half at right angles

Constructing Lines at Right Angles Through Given Points - Given a line and a point, it is possible to construct a line at right angles to the given line that passes through the point.

Constructing Angles Bisectors - Angles Bisectors are lines that cut angles in half

Constructing a 60° Angle - Because equilateral triangles have all angles the same, you can construct a 60° angle in the process. Note: Join up the two sides and you have an equilateral triangle.

Constructing Parallel Lines - Parallel lines are always the same distance apart and never meet - The following method could also be used and is often preferred. - Draw a line. - Set your compass to the required distance - Draw several arcs with the point of the compass along the line. - Using a ruler, join the tops of the arcs to form the parallel line.

Constructing Circles - To construct a circle of a certain radius, set the compass to that length. e.g. Draw a circle with a radius of 2.5 cm 1 2 3 4 5 6 7 Note: If given a diameter, make sure you set the compass to HALF of that value e.g. To draw a circle of diameter 8.4 cm, the compass would be set to 4.2 cm

Constructing Triangles - To construct a triangle given 3 lengths, use the following method: e.g. Draw a triangle of sides 4.8 cm, 4.1 cm and 3.6 cm Step 1: Draw the longest line. Step 2: Set your compass to the other lengths and scribe arcs from each end of the line. Step 3: Join the point where the arcs cross to the ends to form the triangle. 4.1 cm 3.6 cm 4.8 cm Note: Sometimes you may need to use a protractor to draw triangles containing certain angles sizes

Loci 1. Locus of points related to a single point. - A locus is a set of points which fit some geometrical requirements, usually distance There are four main loci to remember: 1. Locus of points related to a single point. e.g. Draw the locus of all points 3 cm from A 3 cm 1. Set your compass to 3 cm 2. Scribe a circle with the compass set on A A 2. Locus of points related to a fixed line. e.g. Draw the locus of all points 2 cm from the infinite line AB e.g. Draw the locus of all points 2 cm from the line segment AB 2 cm 2 cm 2 cm A 2 cm B A B Note: line segments have points at each end

3. Locus of points related two fixed points. - The locus is the perpendicular bisector of the line joining the two points e.g. Show all of the points closer to A than B 1. Join points A and B with a line 2. Construct the perpendicular bisector of the line AB (Line is dashed to show it is not included) A B 3. Shade the side closer to A 4. Locus of points related two fixed lines. - The locus of all points is the pair of angle bisectors e.g. Draw the locus of all points the same distance from both line segments PQ and RS Q R 1. Construct two angle bisectors between both lines S P

3D Shapes Important solids you should be aware of. CUBE CUBOID SPHERE (Rectangular Prism) PYRAMID (can have different shaped bases ` i.e. Square based Pyramid) CYLINDER CONE (Circular based Pyramid) (Circular Prism) NOTE: Solids that have identical ends that are parallel are called PRISMS and their names are based on the shapes of their ends.

Faces, Edges and Vertices - Faces are flat surfaces on a solid - Edges are where two flat surfaces meet - Vertices are where three or more edges meet to form a point e.g. For the following solid a) State how many faces there are and name one A B 6 BCGF D C b) State how many edges there are and name one H G 12 EH E F c) State how many vertices there are and name one 8 A

Nets of Solids - A pattern that can be folded up to form a solid shape is called a NET e.g. Draw a net for the following juice box. Note: other variations of the above net will also work To actually make a shape, tabs are needed on every second edge

Isometric Drawing - Isometric drawings are done on isometric paper which is made up of a series of dots. - Vertical lines represent the height and sloping lines represent the horizontal lines e.g. Draw the following shape in isometric view start with this corner

Drawing Side, Top and Front Views - When blocks are viewed face on, they look like single squares e.g. Draw the top, front, left and right side views of the following shape top left right TOP VIEW front LEFT VIEW FRONT VIEW RIGHT VIEW