1. 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
angle
side
The angle between the two sides is often called the included angle.
We use the abbreviation SAS to stand for Side, Angle and Side.

2. Constructing a triangle given SAS
Construct ABC with AB = 6 cm, B = 68°
and BC = 5 cm.
Start by drawing side AB with a ruler. C
5 cm
Use a protractor to mark an angle of 68° from point B.
68° A
6 cm B
Use a ruler to draw a line of 5 cm from B to C.
Join A to C using a ruler to complete the triangle.

3. Constructing a triangle given ASA
How could we construct a triangle given two angles and the length of the side between them?
angle
angle
side
The side between the two angles is often called the included side.
We use the abbreviation ASA to stand for Angle, Side and Angle.

4. Constructing a triangle given ASA
For example, construct ABC with AB = 10 cm, A = 35° and B = 115°. C
Start by drawing side AB with a ruler.
Use a protractor to mark an angle of 35° from point A.
Use a ruler to draw a long line from A.
115°
35°
Ask pupils how we could check that this triangle has been constructed correctly.
Establish that we could measure angle C to verify that it measures 30°.
Remind pupils that construction lines should not be rubbed out.
Use a protractor to mark an angle of 115° from point B. A
10 cm B
Use a ruler to draw a line from B to meet the other line at point C.

5. Constructing a triangle given SSS
How could we construct a triangle given the lengths of three sides?
side
side
side
Hint: We would need to use a pair of compasses.
We use the abbreviation SSS to stand for Side, Side, Side.

6. Constructing a triangle given SSS
Construct ABC with AB = 4 cm, BC = 3 cm and
AC = 5 cm. C
Start by drawing side AB with a ruler.
Open a pair of compasses to a length of 5 cm.
5 cm
3 cm
Put the compass needle at point A and draw an arc above line AB. A
4 cm B
Next, open the compasses out to a length of 3 cm.
Put the compass needle at point B and draw an arc crossing over the other one. This is point C.
Draw lines AC and BC to complete the triangle.

7. Constructing a triangle given RHS
Remember, the longest side in a right-angled triangle is called the hypotenuse.
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
We use the abbreviation RHS to stand for Right angle, Hypotenuse and Side.

8. Constructing a triangle given RHS
Construct ABC with AB = 5 cm, B = 90° and
AC = 7 cm. C
Start by drawing side AB with a ruler.
7 cm
Extend AB and use compasses to construct a perpendicular at point B. A
5 cm B
Open the compasses to 7 cm.
Ask pupils how we know that, if angle B is a right-angle, side AC must be the hypotenuse.
The hypotenuse, the longest side in a right-angled triangle, must always be the side opposite the right angle.
Review the construction of a perpendicular from a point on a line on slide 24.
Place the compass needle on A and draw an arc on the perpendicular.
Label this point C and complete the triangle.

9. Constructing triangles
Use this activity to review methods for constructing triangles given incomplete information.
Ask pupils to draw sketches of the given triangle and to use these sketches to justify whether or not the triangle can be constructed.

10. S6.3 Constructing lines and angles
Contents
S6 Construction and loci A
S6.1 Drawing lines and angles A
S6.2 Constructing triangles A
S6.3 Constructing lines and angles A
S6.4 Constructing nets A
N6.5 Loci

11. Bisecting lines
Two lines bisect each other if each line divides the other into two equal parts.
Line CD bisects line AB at right angles. C A B D
Discuss the meaning of saying that AB and CD are perpendicular.
We indicate equal lengths using dashes on the lines.
When two lines bisect each other at right angles we can join the end points together to form a rhombus.

12. Bisecting angles
A line bisects an angle if it divides it into two equal angles.
In the diagram below line BD bisects ABC. A B C D

13. The perpendicular bisector of a line
We can construct the perpendicular bisector of a line segment AB using a ruler and a pair of compasses as follows.
Place the compass needle at point A and open them to more than half the length of the line. A B
Draw an arc.
Move the compass needle to point B and draw another arc.
Ask pupils how we could add lines to this diagram to make a rhombus.
The orange line, the perpendicular bisector, forms the set of points, or locus, of the points that are equidistant from points A and B.
Remind pupils that construction lines should not be rubbed out.
Join the points where the arcs cross to form the perpendicular bisector.

14. The bisector of an angle
We can construct the bisector of an angle ABC using a ruler and a pair of compasses as follows. A
Place the compass needle at point B and draw an arc that cuts lines AB and BC. P R
Place the compass needle at P and draw an arc.
Repeat this with the compass needle at Q.
When the diagram is complete ask pupils to name the two equal angles. These are angle ABR and angle RBC. B Q C
Join point B to the point where the two arcs cross to make the angle bisector.

15. The perpendicular from a point to a line
We can construct the perpendicular from point P to line segment AB using a ruler and pair of compasses as follows.
Start by placing the compass needle at point P and drawing an arc crossing the line segment AB at points Q and R. P Q R
Place the of the compass needle at point Q and draw an arc below the line. A B
Repeat this at point R.
Join point P to the point where the arcs cross to form the perpendicular from point P to line segment AB.

16. 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 pair of compasses as follows.
Start by placing the compass needle at point P and use it to draw an arc crossing the line segment AB at points Q and R. Q P R A B
Open the compasses out a bit more and place the point at Q to draw an arc.
Repeat this at point R without adjusting the compasses.
Join the points where the arcs cross to form the perpendicular through point P and line segment AB.

17. S6 Construction and loci
Contents
S6 Construction and loci A
S6.1 Drawing lines and angles A
S6.2 Constructing triangles A
S6.3 Constructing lines and angles A
S6.4 Constructing nets A
N6.5 Loci

18. Imagining paths
A locus is a set of points that satisfy a rule or set of rules.
The plural of locus is loci.
We can think of a locus as a path traced out by a moving point.
For example:
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.
Ask pupils to describe the path of the golf ball by sketching it or tracing the path in the air with their fingers.

19. 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.

20. 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?
Ask pupils to trace out the path using their finger tips or by sketching it.
How could you represent the path in two dimensions?
What about in three dimensions?

21. Imagining paths
A nervous woman paces up and down in one of the capsules on the Millennium Eye as she ‘enjoys’ the view.
Ask pupils to trace out the path using their finger tips or by sketching it.
Can you imagine the path traced out by the woman?
How could you represent the path in two dimensions?
What about in three dimensions?

22. 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.
Ask pupils to trace out the path using their finger tips or by sketching it.
Can you imagine the path traced out by the man?
How could you represent the path in two dimensions?
What about in three dimensions?

23. 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?
Ask pupils to trace out the path using their finger tips or by sketching it.
In two dimensions the locus would be a circle.
In three dimensions the locus would be the surface of a sphere.
How could you represent the path in two dimensions?
What about in three dimensions?

24. Walking turtle
Introduce the locus of points that are a fixed distance from a point, the locus of points that are equidistant from two fixed points and the locus of points that are a fixed distance from a line by walking the turtle along each path.
Change the positions of the fence and the plants and ask a volunteer to come to the board and sketch a required path using the pen tool (set to draw straight lines if necessary). Animate the turtle and compare the paths.

25. The locus of points from a fixed point
Imagine placing counters so that their centres are always 5 cm from a fixed point P.
5 cm P
Tell pupils that some loci are defined in exact mathematical terms. These loci have to be constructed exactly using compasses and a ruler.
Ask pupils to describe how we could construct the locus of points a fixed distance from a point (using a pair of compasses).
As an extension ask pupils to imagine this locus of points in three dimensions (the surface sphere).
Describe the locus made by the counters.
The locus is a circle with a radius of 5 cm and centre at point P.

26. The locus of points from a line segment
Imagine placing counters so that their centres are always 3 cm from a line segment AB. A B
Ask pupils to describe the locus.
As an extension ask pupils to imagine this locus of points in three dimensions (the surface of a cylinder with a hemi-sphere at each end).
Describe the locus made by the counters.
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.

27. 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 Q
Ask pupils how the locus of the points equidistant from two fixed point can be constructed. Review the use of compasses to construct a perpendicular bisector, if necessary.
As an extension ask pupils to imagine this locus of points in three dimensions (a flat plane bisecting the line joining the two points at right angles).
Describe the locus made by the counters.
The locus is the perpendicular bisector of the line joining the two points.

28. 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.
Ask pupils how the locus of the points equidistant from two lines can be constructed. Review the use of a pair of compasses to construct an angle bisector, if necessary.
Ask pupils what the locus would look like if the two lines were extended back beyond the intersection.
Describe the locus made by the counters.
The locus is the angle bisector of the angle where the two lines intersect.

29. Rolling shapes along lines
Use this activity to demonstrate the locus of points traced out when various shapes are rolled along a straight line.
Ask pupils to describe what they expect the locus to look like before revealing it.
Draw attention to the fact that the more sides the shape has the more the path traced looks like that of a circle.