S9 Construction and loci KS4 Mathematics S9 Construction and loci

S9.1 Constructing triangles Contents S9 Construction and loci A S9.1 Constructing triangles A S9.2 Geometrical constructions A S9.3 Imagining paths and regions A S9.4 Loci A S9.5 Combining loci

Equipment needed for constructions 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 Constructions should be drawn in pencil and construction lines should not be rubbed out.

Constructing triangles To accurately construct a triangle you need to know: To accurately construct a triangle you need to know: The length of two sides and the included angle (SAS) The size of two angles and a side (ASA) The lengths of all three sides (SSS) or A right angle, the length of the hypotenuse and the length of one other side (RHS)

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 Discuss how this could be constructed using a ruler and a protractor. 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

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 Point out that if we are given two angles in a triangle we can work out the size of the third angle using the fact that the angles in a triangle add up to 180°. If the side we are given is not the included side then we can work out the third angle to make the side we know an included side. Discuss how this triangle could be constructed using a ruler and a protractor. The side between the two angles is often called the included side. We use the abbreviation ASA to stand for Angle, Side and Angle.

Constructing a triangle given ASA 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.

Constructing a triangle given SSS How could we construct a triangle given the lengths of three sides? side side side Discuss how this triangle could be constructed using a ruler and compasses. Hint: We would need to use a pair of compasses. We use the abbreviation SSS to stand for Side, Side, Side.

Constructing a triangle given SSS

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.

Constructing a triangle given RHS Ask pupils how we know that if angle B is the right-angle side AC must be the hypotenuse. Stress that 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 19.

S9.2 Geometrical constructions Contents S9 Construction and loci A S9.1 Constructing triangles A S9.2 Geometrical constructions A S9.3 Imagining paths and regions A S9.4 Loci A S9.5 Combining 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. C A B D State that lines 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.

Bisecting angles A line bisects an angle if it divides it into two equal angles. For example, in this diagram line BD bisects ABC. A B C D

The perpendicular bisector of a line 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.

The bisector of an angle When the diagram is complete ask pupils to name the two equal angles. These are angle ABR and angle RBC.

The perpendicular from a point to a line

The perpendicular from a point on a line

S9.3 Imagining paths and regions Contents S9 Construction and loci A S9.1 Constructing triangles A S9.2 Geometrical constructions A S9.3 Imagining paths and regions A S9.4 Loci A S9.5 Combining loci

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 or region traced out by a moving point. For example, Imagine the path traced by a football as it is kicked into the air and returns to the ground. Ask pupils to describe the path of the ball by sketching it or tracing the path in the air with their fingers.

Imagining paths The path of the ball as it travels through the air will look something like this: The shape of the path traced out by the ball has a special name. Do you know what it is? Remind pupils that they have met parabola when sketching the graph of x2. Explain that a projectile is acted on by gravity. The physical laws governing the path of a projectile mean that the locus of points traced out will be parabolic. Other factors could alter this path such as air resistance. This shape is called a parabola.

Imagining paths Some fluffy dice hangs from the rear-view mirror in a car and swing from side to side as the car moves forwards. Ask pupils to trace out the path using their finger tips or by sketching it. This path is similar the a sine curve. Can you imagine the path traced out by one of the die? 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. 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?

Franco’s Pizza House is not drawn to scale! Imagining regions Franco promises free delivery for all pizzas within 3 miles of his Pizza House. Franco’s Pizza House is not drawn to scale! 3 miles The region is a circle of radius 3 miles. Can you describe the shape of the region within which Franco can deliver his pizzas free-of-charge?

Grazing sheep Explain that the sheep is tethered to the point P on the edge of the barn by a rope that is 4 m long. Ask pupils to describe the region of grass that the sheep can graze. Press the play button to illustrate the region on the board and show the radius of each arc using the straight pen tool. Explain that as the sheep gets to the corners there is only 1 m of rope left and so the radius of the sector at that corner is 1 m. Change the point from which the rope is attached. Repeat the activity to investigate which position would give the sheep the largest grazing area.

S9 Construction and loci Contents S9 Construction and loci A S9.1 Constructing triangles A S9.2 Geometrical constructions A S9.3 Imagining paths and regions A S9.4 Loci A S9.5 Combining loci

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 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.

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.

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.

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 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.

The locus of points from a given shape Imagine placing counters so that they are always the same distance from the outside of a rectangle. Discuss the fact that the locus is not rectangular but rounded at the corners. We can think of this as a combination of the locus of points that are a fixed distance from a line (around the edges) and the locus of points that are a fixed distance from a point (around the corners). Describe the locus made by the counters. The locus is not rectangular, but is rounded at the corners.

The locus of points from a given shape Discuss the fact that for each shape the locus is parallel to the edges and rounded at the corners. Ask pupils to use the pen tool to sketch what they think the path will look like before starting the animation. Again, we can think of these loci as a combination of the locus of points that are a fixed distance from a line (around the edges) and the locus of points that are a fixed distance from a point (around the corners).

S9 Construction and loci Contents S9 Construction and loci A S9.1 Constructing triangles A S9.2 Geometrical constructions A S9.3 Imagining paths and regions A S9.4 Loci A S9.5 Combining loci

Combining loci 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.

Tethered goats Demonstrate various combinations of grazing areas. Ask pupils how we could accurately construct the required loci using a ruler and compasses. Ask pupils to indicate points that satisfy conditions given by the loci chosen. For example, ask pupils to indicate two points that are both 14 m from P and 14 m from R. Investigate which combinations would give the smallest and largest overlapping areas.

The intersection of two loci Suppose we have a red counter and a blue counter that are 9 cm apart. 6 cm 5 cm Draw an arc of radius 6 cm from the blue counter. Draw an arc of radius 5 cm from the red counter. 5 cm 6 cm 9 cm Explain that when we use counters to represent points we always take measurements from the centre of the counter. Explain that all the points that lie on the first arc are 6 cm from the blue counter and all the points that lie on the second arc lie 5 cm from the red counter. The points where the two arcs cross are both 6 cm from the blue counter and 5 cm from the red counter. Stress to pupils that all construction lines should be left in place and not rubbed out. How can we place a yellow counter so that it is 6 cm from the blue counter and 5 cm from the red counter? There are two possible positions.