© Boardworks Ltd 2004 1 of 41 S6 Construction and loci KS3 Mathematics.

© Boardworks Ltd 20041 of 41

S6 Construction and loci

KS3 Mathematics


S6.1 Drawing lines and angles

Contents


S6.2 Constructing triangles

N6.5 Loci

S6.3 Constructing lines and angles

S6.4 Constructing nets


Equipment needed for constructions

Before you begin make sure you have the following equipment:

A protractorA ruler marked in

cm and mm

A compass A sharp pencil


Drawing lines

To draw a line of given length use a ruler and a sharp pencil.

For example, draw line segment AB of length 56 mm.

Draw a point and label it A.

A

Place your ruler with 0 mm at the point A.

0 1 2 3 4 5 6 7

Mark point B at 56 mm.

Draw line segment AB.

B


Drawing angles

To draw a given angle use a ruler, a protractor and a sharp pencil.

For example, construct ABC = 68°

Start by drawing line segment AB.

Place a protractor so that point B is at the centre of the small circle.

Reading the scale so that 0° is on the line, mark point C at 68°.

Use a ruler to draw a line from point B to point C.A B

C


Drawing reflex angles

To draw a reflex angle we can use a 360° protractor.

To draw a reflex angle using a 180° protractor, start by subtracting the angle from 360°.

For example, to draw an angle of 243° work out 360° – 243°.

360° – 243° = 117°.

Draw an angle of 117°.117°

243°The reflex angle will be the required 243°.


Drawing lines and angles

Draw the following lines and angles as accurately as possible using a ruler and a protractor.

5 cm

4 cm78°

47 mm

128°

62 mm

305°38 mm

54 mm



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N6.5 Loci






Constructing a triangle given SAS

How could we construct a triangle given the lengths of two of its sides and the angle between them?

side

side

angle

The angle between the two sides is often called the included angle.

We use the abbreviation SAS to stand for Side, Angle and Side.


Constructing a triangle given SAS

For example, construct ABC with AB = 6 cm, B = 68° and BC = 5 cm.

Start by drawing side AB with a ruler.

A

Use a protractor to mark an angle of 68° from point B.

6 cmUse a ruler to draw a line of 5 cm from B to C.

Join A to C using a ruler to complete the triangle.

B

C

5 cm

68°


Constructing a triangle given ASA

How could we construct a triangle given two angles and the length of the side between them?

The side between the two angles is often called the included side.

We use the abbreviation ASA to stand for Angle, Side and Angle.

side

angleangle


Constructing a triangle given ASA

For example, construct ABC with AB = 10 cm, A = 35° and B = 115°.


Use a protractor to mark an angle of 35° from point A.

Use a ruler to draw a long line from A.

A B10 cmUse a protractor to mark an angle of 115° from point B.

Use a ruler to draw a line from B to meet the other line at point C.

C

35°115°


Constructing a triangle given SSS

How could we construct a triangle given the lengths of three sides?

side

We use the abbreviation SSS to stand for Side, Side, Side.

side side

Hint: We would need to use a compass.


Constructing a triangle given SSS

For example, construct ABC with AB = 4 cm, BC = 3 cm and AC = 5 cm.


Use a compass and stretch it out to a length of 5 cm.

Put the point of the compass at point A and draw an arc above line AB. A B4 cm

Next, stretch the compass out to a length of 3 cm.

Put the point of the compass at point B and draw an arc crossing over the other one. This is point C.

C

3 cm5 cm

Draw lines AC and BC to complete the triangle.


Constructing a triangle given RHS

Remember, the longest side in a right-angled triangle is called the hypotenuse.

We use the abbreviation RHS to stand for Right angle, Hypotenuse and Side.

How could we construct a right-angled triangle given the right angle, the length of the hypotenuse and the length of one other side?

hypotenuse

right angle

side


Constructing a triangle given RHS

For example, construct ABC with AB = 5 cm, B = 90° and AC = 7 cm.


A

Extend AB and use a compass to construct a perpendicular at point B.

5 cmOpen the compass to 7 cm.

Place the point of the compass on A and draw an arc on the perpendicular.

B

Label this point C and complete the triangle.

C

7 cm


Constructing triangles



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N6.5 Loci






Bisecting lines

Two lines bisect each other if each line divides the other into two equal parts.

For example, line CD bisects line AB at right angles.

We indicate equal lengths using dashes on the lines.

A B

C

D

When two lines bisect each other at right angles we can join the end points together to form a rhombus.


Bisecting angles

A line bisects an angle if it divides it into two equal angles.

A

B C

For example, in this diagram line BD bisects ABC.

D


The perpendicular bisector of a line

We can construct the perpendicular bisector of a line segment AB using a ruler and a compass as follows.

A B

Place the point of a compass at point A and open it so that it is more than half the size of the line.

Draw an arc.

Move the point of the compass to point B and draw an other arc.

Join the points where the arcs cross to form the perpendicular bisector.


The bisector of an angle

We can construct the bisector of an angle ABC using a ruler and a compass as follows.

Place the point of a compass at point B and draw an arc that cuts lines AB and BC.

B C

A

P

Q

Place the point of the compass at point P and draw an arc.

R

Repeat this with the point of the compass at point Q.

Join point B to the point where the two arcs cross to make the angle bisector.


The perpendicular from a point to a line

We can construct the perpendicular from point P to line segment AB using a ruler and a compass as follows.

A B

PStart by placing the point of the compass at point P and use it to draw an arc crossing the line segment AB at points Q and R.

Q R

Place the point of the compass at point Q and draw an arc below the line.

Repeat this at point R.

Join point P to the points where the arcs cross to form the perpendicular from point P to line segment AB.


The perpendicular from a point on a line

We can construct the perpendicular from the point P on line segment AB using a ruler and a compass as follows.

Start by placing the point of the compass at point P and use it to draw an arc crossing the line segment AB at points Q and R.

A B

PQ R

Open the compass out a bit more and place the point at Q to draw an arc.

Repeat this at point R without changing the compass.

Join point the points where the arcs cross to form the perpendicular through point P and line segment AB.



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Constructing nets

Before constructing an accurate net of a solid shape it is sensible to start by making a sketch of how we expect the completed net to look.Remember, in maths, a sketch is a diagram that is not drawn to scale. We should still use a ruler to do this.Suppose we want to construct the net of a cuboid of length 4 cm, width 2 cm and height 1 cm.We can sketch the net on squared paper as follows:

These are tabs.


Construct a cuboid

Construct nets for the following cuboids on plain paper.

3 cm

2 cm

4 cm

5 cm

3 cm1.5 cm


Construct the net of a tetrahedron

Construct the net of a square-based pyramid with base 3 cm by 3 cm and sloping edges of length 4 cm.

3 cm

3 cm

4 cm

Your sketch should look something like this.

3 cm

3 cm

4 cm


N6.5 Loci

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A locus is a set of points that satisfy a rule or set of rules.

The plural of locus is loci.

Imagining paths

Imagine the path of a golf ball as it is driven on to the green.

The ball bounces several times and drops into the hole.

We can think of a locus as a path traced out by a moving point.

For example,


Imagining paths

The path of the ball might look something like this:

The red dotted line shows the locus of the points traced out by the golf ball.


Imagining paths

A helicopter takes off and rises to a height of 30 m.

It accelerates forward and rises to 1000 m where it levels out then continues on its way.

Can you imagine the path traced out by the helicopter?

How could you represent the path in two dimensions?

What about in three dimensions?


Imagining paths

A nervous woman paces up and down in one of the capsules on the Millennium Eye as she ‘enjoys’ the view.

Can you imagine the path traced out by the woman?




Imagining paths

A man has been out for the evening, has locked himself out of his flat and has to climb the spiral fire escape.

Can you imagine the path traced out by the man?




Imagining paths

A spider is hanging motionless from a web.Imagine moving your hand so that the tip of your finger is always 10 cm from the spider.

Can you imagine the path traced out by your finger tip?




Walking turtle


The locus of points from a fixed point

Imagine placing counters so that their centres are always 5 cm from a fixed point P.

P

Describe the locus made by the counters.

The locus is a circle with a radius of 5 cm and centre at point P.

5 cm


The locus of points from a line segment

Imagine placing counters that their centres are always 3 cm from a line segment AB.

A B


The locus is a pair of parallel lines 3 cm either side of AB. The ends of the line AB are fixed points, so we draw semi-circles of radius 3 cm.


The locus of points from two fixed points

Imagine placing counters so that they are always an equal distance from two fixed points P and Q.

P

The locus is the perpendicular bisector of the line joining the two points.

Q



The locus of points from two lines

Imagine placing counters so that they are an equal distance from two straight lines that meet at an angle.

The locus is the angle bisector of the angle where the two lines intersect.



Rolling shapes along lines

© Boardworks Ltd 2004 1 of 41 S6 Construction and loci KS3 Mathematics.

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Transcript of © Boardworks Ltd 2004 1 of 41 S6 Construction and loci KS3 Mathematics.