Constructions and 3D Drawings

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 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. 4.8 cm 4.1 cm 3.6 cm

Loci - 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. A e.g. Draw the locus of all points 3 cm from A 1. Set your compass to 3 cm 2. Scribe a circle with the compass set on A 2. Locus of points related to a fixed line. 3 cm e.g. Draw the locus of all points 2 cm from the infinite line AB AB e.g. Draw the locus of all points 2 cm from the line segment AB AB 2 cm Note: line segments have points at each end 2 cm

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 AB 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) 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 P Q R S 1. Construct two angle bisectors between both lines

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 VIEW RIGHT VIEWFRONT VIEWLEFT VIEW top left front right

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

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. To actually make a shape, tabs are needed on every second edge Note: other variations of the above net will also work

Reflection e.g. Reflect the object in mirror line m. m Properties: - Reflected lines, angles and areas remain the same size - Points on the mirror line are invariant - Sense or orientation is reversed e.g. Draw in the mirror line for the following reflection. A C B A’ B’ C’ A A’ m Note: When fully describing a reflection, a mirror line must be added and discussed in the instructions

Translation - Is the movement of an object, without twisting, in a particular direction - The movement can be described by a vector or a direction e.g. Translate the object ‘A’ by the vector A Properties: - Translated lines, angles and areas remain the same size A e.g. By what vector has the following object been translated? A’ A’ Vector = Note: When fully describing a translation, a vector or direction must be added and discussed in the instructions

Rotation - To perform rotations a centre of rotation and angle of rotation are needed - If not stated, the direction of a rotation is always anti-clockwise e.g. Rotate the object 90° about X A BC D X A’B’ C’D’ Properties: - 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 e.g. Find the centre of rotation A A’ X 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 Note: When fully describing a rotation, a centre, angle and direction must be added and discussed in the instructions

Enlargement - To perform an enlargement, a centre and scale factor are needed - An object can shrink or enlarge e.g. Enlarge the object by scale factor of 2 and centre X A X A’ e.g. Enlarge the object by scale factor of -0.5 and centre X Note: Negative scale factors mean the image ends up on the other side of the centre and reversed X Note: When fully describing an enlargement, a centre and scale factor must be added and discussed in the instructions A A’

Enlargement Continued - To calculate the scale factor, use the following formula Scale factor (s.f.) = length of image length of object length of object e.g. Calculate the scale factor of the following enlargement A CB A’ B’C’ Scale factor (s.f.) = = The centre of enlargement is at the intersection of the lines joining points to their images e.g. Find the centre and scale factor of the enlargement X Scale factor (s.f.) = 1 2 AA’ Centre = X