12 CONSTRUCTIONS AND LOCI

Transcription

1 12 ONSTRUTIONS N LOI rchitects make scale drawings of projects they are working on for both planning and presentation purposes. Originally these were done on paper using ink, and copies had to be made laboriously by hand. Later they were done on tracing paper so that copying was easier. omputer-generated drawings have now largely taken over, but, for many of the top architecture firms, these too have been replaced, by architectural animation. Objectives In this chapter you will: use a ruler and a pair of compasses to draw triangles given the lengths of the sides use a straight edge and a pair of compasses to construct perpendiculars and bisectors construct and bisect angles using a pair of compasses draw loci and regions learn how to draw, use and interpret scale drawings. 194 efore you start You need to: be able to make accurate drawings of triangles and 2 shapes using a ruler and a protractor be able to draw parallel lines using a protractor and ruler have some understanding of ratio be able to change from one metric unit of length to another.

2 12.1 onstructing triangles 12.1 onstructing triangles Objective You can draw a triangle when given the lengths of its sides. Why do this? If you were redesigning a garden and wanted a triangular border you would need to make a plan first and draw the triangles accurately. Get Ready 1. Use a ruler and protractor to make an accurate drawing of this triangle. Measure, and angle cm Key oints Two triangles are congruent if they have exactly the same shape and size. One of four conditions must be true for two triangles to be congruent: SSS, SS, S and RHS (see Section 8.1). onstructing a triangle using any one of these sets of information therefore creates a unique triangle. More than one possible triangle can be created from other sets of information. Example 1 Make an accurate drawing of the triangle shown in the sketch. Watch Out! 4 cm 5 cm 6 cm The diagram in the question will not be drawn accurately so don t measure it, use the dimensions marked. Start by drawing the base 5 cm long. Label the ends and. raw an arc, radius 4 cm. centre. raw an arc, radius 6 cm, centre. is the point where the two arcs intersect. is 4 cm from and 6 cm from. Join to and to. 195

3 hapter 12 onstructions and loci Example 2 Show that there are two possible triangles in which 5.6 cm, 3.3 cm and angle 31. raw the line with length 5.6 cm. Using a protractor, draw an angle of 31 at raw an arc of 3.3 cm from point, to locate the possible positions of. Triangle 1 and 2 both have the given measurements. 31 Exercise 12 Questions in this chapter are targeted at the grades indicated. 1 Here is a sketch of triangle YZ. onstruct triangle YZ. Z 6 cm 7.5 cm Y 4.5 cm 2 onstruct an equilateral triangle with sides of length 5 cm. 3 onstruct the triangle YZ with sides Y 4.2 cm, YZ 5.8 cm and Z 7.5 cm. 4 Here is a sketch of the quadrilateral EF. Make an accurate drawing of quadrilateral EF. F 3.5 cm E 4 cm 5 cm 4.5 cm 6 cm 5 The rhombus KLMN has sides of length 5 cm. The diagonal KM 6 cm. Make an accurate drawing of the rhombus KLMN. 6 Explain why it is not possible to construct a triangle with sides of length 4 cm, 3 cm and 8 cm. 196

4 12.2 erpendicular lines 12.2 erpendicular lines Objective You can construct perpendicular lines using a straight edge and compasses. Why do this? Many structures involve lines or planes that are perpendicular, for example the walls and floor of a house are perpendicular. Get Ready 1. raw a circle with a radius of 4 cm. 2. Mark two points and 6 cm apart. Mark the points that are 5 cm from and 5 cm from. 3. raw two straight lines which are perpendicular to each other. Key oints bisector cuts something exactly in half. perpendicular bisector is at right angles to the line it is cutting. You can use a straight edge and compass in the construction of the following: the perpendicular bisector of a line segment the perpendicular to a line segment from a point on it the perpendicular to a line segment from a point not on the line. Example 3 onstruct the perpendicular bisector of the line. raw arcs centred on above and below the line with a radius more than half of. raw arcs centred on above and below the line with the same radius as before. Join to. is the required perpendicular bisector. Example 4 onstruct the perpendicular to a point on a line. Y raw arcs with the same radius centre to cut at and Y. Y onstruct the perpendicular bisector of Y. Since Y, this must go through. bisector perpendicular bisector construction line segment 197

5 hapter 12 onstructions and loci Example 5 onstruct the perpendicular to a line from a point not on the line. Y Start by drawing arcs with the same radius, centre to cut the line (extended if necessary) at and Y. Y Then construct the perpendicular bisector of Y. Exercise 12 1 raw line segments of length 10 cm and 8 cm. Using a straight edge and a pair of compasses, construct the perpendicular bisector of each of these line segments. 2 raw these lines accurately, and then construct the perpendicular from the point. a b 3 cm 7 cm 2 cm 9 cm 3 raw a line segment, a point above it,, and a point below it, Q. onstruct the perpendicular from to, and from Q to onstructing and bisecting angles Objectives You can construct certain angles using compasses. You can construct the bisector of an angle using a straight edge and compasses. You can construct a regular hexagon inside a circle. Why do this? You may need to bisect an angle accurately when cutting a tile to place in an awkward corner. Get Ready 1. raw a circle with a radius of 3 cm. 2. raw an angle of Use a protractor to bisect an angle of

6 12.3 onstructing and bisecting angles Example 6 onstruct an angle of a 60 b 120. a Start by drawing an arc at the point. Where the arc cuts the line, label the point. Keep your compasses the same width and put the point at. raw an arc to cut the first one. 60 Join up to get a 60º angle. Label the point. is an equilateral triangle. b raw a longer first arc and then draw a third arc from point with the same radius. 120 Example 7 onstruct the bisector of the angle. Start by drawing arcs with the same radius (or a single arc), centre to cut the arms and at and Y. Y Then draw an arc centre radius and an arc centre Y radius to cross at. Join to to get the angle bisector. Y angle bisector 199

7 hapter 12 onstructions and loci Example 8 onstruct a regular hexagon inside a circle. raw a circle and mark a point on its circumference. Keep the compasses set at the size of the radius, and from point draw an arc that cuts the circle at point. Repeat the process until six points are marked on the circumference. Join the points to make a regular hexagon. Exercise 12 1 opy the diagrams and construct the bisector of the angle. a b 200

8 12.4 Loci 2 opy the diagrams and construct the bisector of angle Q in the triangle QR. a R b Q R Q 3 onstruct each of the following angles. a 60 b 120 c 90 d 30 e 45 4 raw a regular hexagon in a circle of radius 4 cm. 5 raw a regular octagon in a circle of radius 4 cm Loci Objective You can draw the locus of a point. Why do this? Scientists studying interference effects of radio waves need to plot paths that are equidistant from two or more transmitters. They use loci to do this. Get Ready 1. ut a cross in your book. Mark some points which are 3 cm from the cross. 2. ut two crosses and less than 3 cm apart in your book. Mark points which are 3 cm from each cross. 3. raw two parallel lines. Mark any points which are the same distance from both lines. Key oints locus is a line or curve, formed by points that all satisfy a certain condition. locus can be drawn such that: its distance from a fixed point is constant it is equidistant from two given points its distance from a given line is constant it is equidistant from two lines. Example 9 Show the locus of all points which are at a distance of 3 cm from the fixed point O. 3 cm The locus is a circle, radius 3 cm, centre O. ll the points on the circle are 3 cm from O. O locus equidistant 201

9 hapter 12 onstructions and loci Example 10 Show the locus of all points which are equidistant from the points and Y. Examiner s Tip Y onstruct the perpendicular bisector of the line Y. raw loci accurately. Use a pair of compasses, a ruler and a straight edge. ll points on the perpendicular bisector are equidistant (the same distance) from and Y. Example 11 Show the locus of all points which are 3 cm from the line segment Y. Y 3 cm These lines are parallel to Y and 3 cm away from it. ll points on the semicircles are 3 cm from the point Y. Exercise 12 1 Mark two points and approximately 6 cm apart. raw the locus of all points that are equidistant from and. 2 raw the locus of all points which are 3.5 cm from a point. 3 raw the locus of a point that moves so that it is always 1.5 cm from a line 5 cm long. 4 raw two lines Q and QR, so that the angle QR is acute. raw the locus of all points that are equidistant between the two lines Q and QR. 202

10 12.5 Regions 12.5 Regions Objective You can draw regions. Why do this? If you tether a goat to a point in your garden to eat the grass, you might want to check that the region it can access doesn t include the flowerbed. Get Ready 1. ut a cross in your book. Mark some points which are less than 3 cm from the cross. 2. ut two crosses and in your book. Mark points which are closer to than to. 3. raw two parallel lines. Mark any points which are further from one line than the other. Key oints set of points can lie inside a region rather than on a line or curve. The region of points can be drawn such that: the points are greater than or less than a given distance from a fixed point the points are closer to one given point than to another given point the points are closer to one given line than to another given line. Example 12 raw the region of points which are less than 2 cm from the point O. The locus is a circle, radius 2 cm, centre O. ll the points on the circle are 2 cm from O. O Example 13 raw the region of all points which are closer to the point than to the point Y. Y ll the points to the left of the perpendicular bisector of Y are closer to than to Y. region 203

11 hapter 12 onstructions and loci Example 14 is a square of side 4 cm. raw the region of points inside the rectangle that are both more than 3 cm from point and more than 2 cm from the line. 4 cm 4 cm 3 cm Find the locus of points 3 cm from point inside the square. Find the locus of points 2 cm from the line inside the square. 2 cm Shade the area that is both more than 3 cm from point and more than 2 cm from the line. Exercise 12E 1 Shade the region of points which are less than 2 cm from a point. 2 Shade the region of points which are less than 2.6 cm from a line 4 cm long. 3 Mark two points, G and H, roughly 3 cm apart. Shade the region of points which are closer to G than to H. 4 raw two lines E and EF, so that the angle EF is acute. Shade the region of points which are closer to EF than to E. 5 aby Tommy is placed inside a rectangular playpen measuring 1.4 m by 0.8 m. He can reach 25 cm outside the playpen. Show the region of points Tommy can reach beyond the edge of the playpen. 204

12 12.6 Scale drawings and maps 12.6 Scale drawings and maps Objectives You can read and construct scale drawings. You can draw lines and shapes to scale and estimate lengths on scale drawings. You can work out lengths using a scale factor. Why do this? When a new aeroplane is being designed or an extension to a house is planned, accurate scale drawings have to be made. Get Ready 1. onvert from cm to km: 2. onvert from km to cm: a cm b cm. a 4 km b 0.3 km. Key oints Here is a picture of a scale model of a Saturn rocket. The model has been built to a scale of 1 : 24. This means that every length on the model is shorter than the length on the real rocket, with a length of 1 cm on the model representing a length of 24 cm on the real rocket. The real rocket is an enlargement of the model with a scale factor of 24; the model is a smaller version of the real rocket with a scale factor of In general, a scale of 1 : n means that: a length on the real object the length on the scale diagram or model n a length on the scale drawing or model the length on the real object n. Example 15 The Empire State uilding is 443 m tall. ill has a model of the building that is 88.6 cm tall. a alculate the scale of the model. Give your answer in the form 1 : n. b The pinnacle at the top of ill s model is 12.4 cm in length. Work out the actual length of the pinnacle at the top of the Empire State uilding. Give your answer in metres. a Height of building cm Scale factor Scale of model 1 : 500 oth heights have to be in the same units. hange 443 m to cm by multiplying by 100. Height of building Scale factor Height of model b Length of pinnacle on building cm Length of pinnacle on building m Length on model Length on building 500. Length on building Length on model 500. hange cm to m by dividing by 100. scale factor scale diagram 205

13 hapter 12 onstructions and loci Example 16 The scale of a map is 1 : a On the map, the distance between two churches is 6 cm. Work out the real distance between the churches. Give your answer in kilometres. b The real distance between two train stations is 12 km. Work out the distance between the two train stations on the map. Give your answer in centimetres. Method 1 a Real distance between churches cm 3000 m 3 km hange cm to m, divide by 100. hange m to km, divide by scale of 1 : means: real distance map distance b 12 km cm hange km to cm by multiplying by istance between stations on map cm Map distance real distance Method 2 Map distance of 1 cm represents real distance of 0.5 km. a 6 cm on the map represents real distance of km. istance between the churches 3 km. 1 : means 1 cm : cm or 1 cm : 500 m or 1 cm : 0.5 km b Real distance of 12 km represents map distance of cm. istance between the stations on map 24 cm. Exercise 12F O2 O3 1 This is an accurate map of a desert island. There is treasure buried on the island at T. Key to map palm trees R rocks cliffs T treasure The real distance between the palm trees and the cliffs is 5 km. a Find the scale of the map. Give your answer in the form 1 cm represents n km, giving the value of n. b Find the real distance of the treasure from: i the cliffs ii the palm trees iii the rocks. T R 206

14 12.6 Scale drawings and maps 2 On a map of England, 1 cm represents 10 km. a The distance between Hull and Manchester is 135 km. Work out the distance between Hull and Manchester on the map. b On the map, the distance between London and York is 31.2 cm. Work out the real distance between London and York. 3 Here is part of a map, not accurately drawn, showing three towns: lphaville (), eecombe () and eeton (). a Using a scale of 1 : , accurately draw this part of the map. b Find the real distance, in km, between eecombe and eeton. c Use the scaled drawing to measure the bearing of eeton from eecombe. 12 km 8 km N O2 4 This is a sketch of rfan s bedroom. It is not drawn to scale. 4 m raw an accurate scale drawing on cm squared paper of rfan s bedroom. Use a scale of 1 : m 1 m 3 m 3 m 2.5 m 5 space shuttle has a length of 24 m. model of the space shuttle has a length of 48 cm. a Find, in the form 1 : n, the scale of the model. b The height of the space shuttle is 5 m. Work out the height of the model. 6 The distance between ristol and Hull is 330 km. On a map, the distance between ristol and Hull is 6.6 cm. a Find, as a ratio, the scale of the map. b The distance between ristol and London is 183 km. Work out the distance between ristol and London on the map. Give your answer in centimetres. hapter review Two triangles are congruent if they have exactly the same shape and size. One of four conditions must be true for two triangles to be congruent: SSS, SS, S and RHS. onstructing a triangle using any one of these sets of information therefore creates a unique triangle. More than one possible triangle can be created from other sets of information. bisector cuts something exactly in half. perpendicular bisector is at right angles to the line it is cutting. locus is a line or curve, formed by points that all satisfy a certain condition. locus can be drawn such that its distance from a fixed point is constant it is equidistant from two given points its distance from a given line is constant it is equidistant from two lines. 207

15 hapter 12 onstructions and loci set of points can lie inside a region rather than on a line or curve. region of points can be drawn such that: the points are greater than or less than a given distance from a fixed point the points are closer to one given point than to another given point the points are closer to one given line than to another given line. scale of 1 : n means that: a length on the real object the length on the scale diagram or model n a length on the scale drawing or model the length on the real object n. Review exercise 1 8 cm. 6 cm. ngle 52. Make an accurate drawing of triangle. 6 cm iagram NOT accurately drawn 52 8 cm Nov Make an accurate drawing of the quadrilateral. 5 cm iagram NOT accurately drawn cm 8 cm 3 Make an accurate drawing of triangle. iagram NOT accurately drawn cm May Make an accurate drawing of triangle QR. iagram NOT accurately drawn 7.3 cm 13.9 cm Q 8.7 cm R 5 model of the Eiffel Tower is made to a scale of 2 millimetres to 1 metre. The width of the base of the real Eiffel Tower is 125 metres. a Work out the width of the base of the model. Give your answer in millimetres. The height of the model is 648 millimetres. b Work out the height of the real Eiffel Tower. Give your answer in metres June 2008, adapted 208

16 hapter review 6 eeham is 10 km from lston. orting is 20 km from eeham. eetown is 45 km from lston. The diagram below shows the straight road from lston to eetown. This diagram has been drawn accurately using a scale of 1 cm to represent 5 km. lston eetown On a copy of the diagram, mark accurately with crosses (x), the positions of eeham and orting. Nov is a triangle. opy the triangle accurately and shade the region inside the triangle which is both less than 4 centimetres from the point and closer to the line than the line. June 2009, adapted 8 On a copy of the diagram, use a ruler and pair of compasses to construct an angle of 30 at. You must show all your construction lines. Exam Question Report 79% of students answered this question poorly because they did not use two different constructions. Nov 2007, adapted 9 a Mark the points and approximately 8 cm apart. raw the locus of all points that are equidistant from and. d raw the locus of a point that moves so that it is always 3 cm from a line 4.5 cm long. 10 is 5 km north of. 4 km is 4 km from. N is 7 km from. iagram NOT a Make an accurate scale drawing of triangle. 5 km 7 km accurately drawn Use a scale of 1 cm to 1 km. b From your accurate scale drawing, measure the bearing of from. c Find the bearing of from. Nov

17 hapter 12 onstructions and loci 11 On an accurate copy of the diagram use a ruler and pair of compasses to construct the bisector of angle. You must show all your construction lines. Nov 2008, adapted O3 12 is a rectangle. Make an accurate drawing of. Shade the set of points inside the rectangle which are both more than 1.2 centimetres from the point and more than 1 centimetre from the line. 13 raw a line segment 7 cm long. onstruct the perpendicular bisector of the line segment. 14 raw a line segment ST and a point above it, M. onstruct the perpendicular from M to ST. O3 15 s a bicycle moves along a flat road, draw the locus of: a the yellow dot b the green dot. O3 16 raw the locus of a man s head as the ladder he is on slips down a wall. 210