04/12/2018 Drawing in maths – KEY WORDS Sketch Does not need to be accurate. Just needs to show the key features (rough shape, dimensions or measurements labelled). No need to use ruler or other equipment. Plot Usually on a graph or chart, using given coordinates or data. Must be accurate according to the scale given. Construct Must be absolutely accurate. All lengths and angles must be measured out perfectly to within 2mm or 2 degrees. Must use correct equipment to get that accuracy.

CONSTRUCTING TRIANGLES 04/12/2018 CONSTRUCTING TRIANGLES

A ruler marked in cm and mm 04/12/2018 Before you begin make sure you have the following equipment: A ruler marked in cm and mm A protractor A pair of compasses A sharp pencil

To construct a triangle, given 1 side and 2 angles. 04/12/2018 Example To construct a triangle, given 1 side and 2 angles. Construct a triangle of side, 9 cm with angles of 35o and 65o. 1. Draw a straight line 9 cm long. 2. Use a protractor to draw angles of 35o and 65o on either end of line. 3. Draw straight lines from A and B to point of intersection to form the triangle. A B 9 cm

Don’t forget...leave your construction lines visible 04/12/2018 Use this method to construct the following triangles... Side 6cm, angles 45° and 30° Side 5cm, angles 80° and 25° Side 8cm, angles 15° and 70° Side 10cm, angles 105° and 50° Side 4cm, angles 25° and 125° Don’t forget...leave your construction lines visible

To construct a triangle, given 2 sides and 1 angle. 04/12/2018 Example To construct a triangle, given 2 sides and 1 angle. Construct a triangle of sides, 10 cm and 7cm with an angle of 115o. 1. Draw a straight line 10 cm long. 2. Use a protractor to draw an angle of 115o on either end of line. 7 cm 3. Mark off a length of 7 cm. 4. Join end points to form the required triangle. 115o A B 10 cm

Don’t forget...leave your construction lines visible 04/12/2018 Use this method to construct the following triangles... Sides 4cm, 7cm and angle 65° Sides 5cm, 8cm and angle 50° Sides 8cm, 3cm and angle 85° Sides 10cm, 9cm and angle 100° Sides 6cm, 5cm and angle 130° Don’t forget...leave your construction lines visible

To construct a triangle, given 3 sides. 04/12/2018 Example To construct a triangle, given 3 sides. Construct a triangle of sides 7 cm, 3½ cm and 10 cm. 1. Using the longest side as the base, draw a straight line 10 cm long. 2. Set compass to 7 cm, place at A and draw an arc. 3. Set compass to 3½ cm, place at B and draw an arc to intersect the first one. 4. Draw straight lines from A and B to point of intersection. 7 cm 3½ cm A B 10 cm

Don’t forget...leave your construction lines visible 04/12/2018 Use this method to construct the following triangles... Sides 4cm, 7cm and 8cm Sides 5cm, 8cm and 10cm Sides 8.5cm, 3cm and 6.5cm Sides 8.4cm, 9.6cm and 7.1cm Equilateral with sides 6cm Isosceles with sides 6.5, 6.5 and 4 Don’t forget...leave your construction lines visible

Triangle construction revision 04/12/2018 Triangle construction revision Constructing from 1 side, 2 angles (ASA) 1) 45°, 50°, 7cm 2) 30°, 70°, 3cm 3) 70°, 110°, 6.5cm 4) 53°, 20°, 6cm 5) 120°, 42°, 12.5cm 6) 135°, 12°, 9cm Constructing from 2 sides, 1 angle (SAS) 1) 45°, 4cm, 7cm 2) 30°, 6cm, 3cm 3) 70°, 4.5cm, 6.5cm 4) 53°, 8cm, 11cm 5) 120°, 4cm, 12.5cm 6) 135°, 9cm, 9cm Constructing from 3 sides (SSS) 1) 8cm, 10cm, 6cm 2) 7cm, 4cm, 9cm 3) 11cm, 6cm, 8cm 4) 2.5cm, 3cm, 4.5cm 5) 6.2cm, 4.5cm, 7.3cm 6) 9cm, 10cm, 3.6cm

04/12/2018 A B 8 cm 7 cm 6 cm Explain why a triangle with sides 2 cm,10 cm and 7 cm cannot be constructed. A B 9 cm 7 cm 4 cm Under what conditions can it be guaranteed that a triangle can be constructed from 3 given lengths? A B 10 cm 7 cm 3½ cm The sum of the lengths of the smaller sides must exceed the length of the longest side.

CONSTRUCTING BISECTORS 04/12/2018 CONSTRUCTING BISECTORS

04/12/2018 Sometimes you will have to construct the perpendicular bisector to a line Perpendicular means... Bisector means...

Don’t forget...leave your construction lines visible 04/12/2018 Step by step... 1: Draw a line 2: Open your compasses to about 3 4 of the way along your line 3: Draw arcs above and below, with your compass point at one end of the line 4: Repeat at the other end of the line WITHOUT changing the compass setting 5: Join where the arcs cross using a pencil and ruler...this is the perpendicular bisector Don’t forget...leave your construction lines visible

04/12/2018 Constructing a perpendicular bisector x 1) Place your compasses at one end of the line and draw an arc just over half way. 2) Now place the compasses at the other end of the line WITHOUT adjusting the length. Draw another arc. 3) Finally, use your ruler to draw a straight line through the two intersects of the arcs. x

Don’t forget...leave your construction lines visible 04/12/2018 Task Now use this method to construct the perpendicular bisector of the following line lengths.... Length 6.5 cm Length 9.8 cm Length 10.7 cm Length 43 mm Length 27 mm Don’t forget...leave your construction lines visible

04/12/2018 Constructing an angle bisector x 1) Place your compasses at the base of the angle and draw an arc about half way, intersecting both lines 2) Now place the compasses at a point where the arc intersects the line and draw another arc above the original arc 3) Do exactly the same as step 3 on the other intersect WITHOUT adjusting the length of the compasses. 4) Finally draw a line through the intersect of the mini arc and the base of the angle. x

Don’t forget...leave your construction lines visible 04/12/2018 Task Draw and bisect the following angles.... 83° 104° 73° 19° 99° 65° Don’t forget...leave your construction lines visible

OTHER WAYS TO USE CONSTRUCTION 04/12/2018 OTHER WAYS TO USE CONSTRUCTION

To construct a Regular Hexagon. 04/12/2018 To construct a Regular Hexagon. 1. Draw a circle of any radius. 2. With compass fixed at 1 radius place anywhere on the circumference and mark off 6 arcs. 3. Join intersections of arcs together to form a regular hexagon.

04/12/2018 LOCI BASICS

Shade the points less than 4cm from P 04/12/2018 Shade the points less than 4cm from P A circle p 4 cm

04/12/2018 Draw a line showing the points an equal distance from A and B p1 p2 The perpendicular bisector of the line joining the points. 3. Place compass at B, with same distance set and draw 2 arcs to intersect first two. 2. Place compass at A, set over halfway and draw 2 arcs 4. Draw the perpendicular bisector through the points of intersection. 1. Join both points with a straight line. A B

04/12/2018 Draw the locus of points 4cm from AB Place compass on ends of line and draw semi-circles of radii 4cm. Draw 2 lines parallel to AB of equal length and 4cm from it. 4cm

04/12/2018 Shade the region that is more than 2cm away from AB B A C D

Shade the points closer to AB than AC 04/12/2018 Shade the points closer to AB than AC A B C 1. Place compass at A and draw an arc crossing both arms. 3. Draw straight line from A through point of intersection for angle bisector. 2. Place compass on each intersection and set at a fixed distance. Then draw 2 arcs that intersect. The angle bisector A B C

TWO POINTS → perpendicular bisector ONE LINE → running track or line 04/12/2018 The four loci you need to know Decide for each statement in the question what it is referring to out of these four options, and draw the appropriate locus… ONE POINT → circle TWO POINTS → perpendicular bisector ONE LINE → running track or line TWO LINES → angle bisector

04/12/2018 COMBINING LOCI

04/12/2018 Suppose two goats, Archimedes and Babbage, occupy a fenced rectangular area of grass of length 18 m and width 12 m. Archimedes is tethered so that he can only eat grass that is within 12 m from the fence PQ and Babbage is tethered so that he can only eat grass that is within 14 m of post R. Describe how we could find the area that both goats can graze.

04/12/2018 Combining loci Look for the words “shade the region” to know it is a loci question For each of the statements given, decide if it is referring to 1 point, 2 points, 1 line or 2 lines, and so which construction you need to be doing If there is a scale given, use that carefully to decide how big to make any circle or running track Once all loci are complete, re-read the statements and use them to decide which region needs to be shaded

INSIDE the blue circle… 04/12/2018 BELOW the pink line… INSIDE the blue circle… 2 points – perpendicular bisector 1 point – circle

04/12/2018 12cm 8cm

04/12/2018 5cm 3.5cm

04/12/2018 12cm 4cm 4cm 10cm

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