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© Boardworks Ltd 2005 of 37 S9 Construction and loci KS4 Mathematics

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KS4 Mathematics. S9 Construction and loci. S9 Construction and loci. Contents. A. S9.2 Geometrical constructions. A. S9.1 Constructing triangles. S9.3 Imagining paths and regions. A. S9.4 Loci. A. S9.5 Combining loci. A. Equipment needed for constructions. - PowerPoint PPT PresentationTRANSCRIPT

S9 Construction and lociS9.4 Loci

Before you begin make sure you have the following equipment:

A protractor

A pair of compasses

A sharp pencil

Constructions should be drawn in pencil and construction lines should not be rubbed out.

© Boardworks Ltd 2005

or

The lengths of all three sides (SSS)

The length of two sides and the included angle (SAS)

A right angle, the length of the hypotenuse and the length of one other side (RHS)

© Boardworks Ltd 2005

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.

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

© Boardworks Ltd 2005

1813.unknown

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

angle

angle

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.

© Boardworks Ltd 2005

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.

1814.unknown

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 pair of compasses.

Discuss how this triangle could be constructed using a ruler and compasses.

© Boardworks Ltd 2005

1815.unknown

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

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.

857.unknown

S9.5 Combining loci

S9.1 Constructing triangles

* of 37

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.

State that lines AB and CD are perpendicular.

© Boardworks Ltd 2005

* of 37

Bisecting angles

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

D

A

B

C

© Boardworks Ltd 2005

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.

1816.unknown

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.

1817.unknown

1818.unknown

1819.unknown

A

A

A

A

A

* of 37

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 traced by a football as it is kicked into the air and returns to the ground.

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

For example,

Ask pupils to describe the path of the ball by sketching it or tracing the path in the air with their fingers.

© Boardworks Ltd 2005

* of 37

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?

This shape is called a parabola.

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.

© Boardworks Ltd 2005

* of 37

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.

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?

Ask pupils to trace out the path using their finger tips or by sketching it. This path is similar the a sine curve.

© Boardworks Ltd 2005

* of 37

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?

How could you represent the path in two dimensions?

What about in three dimensions?

Ask pupils to trace out the path using their finger tips or by sketching it.

© Boardworks Ltd 2005

* of 37

Imagining regions

Franco promises free delivery for all pizzas within 3 miles of his Pizza House.

Can you describe the shape of the region within which Franco can deliver his pizzas free-of-charge?

3 miles

© Boardworks Ltd 2005

* of 37

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.

863.unknown

© Boardworks Ltd 2005

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

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).

© Boardworks Ltd 2005

Imagine placing counters so that their centres are always

3 cm from a line segment AB.

A

B

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.

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).

© Boardworks Ltd 2005

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

Describe the locus made by the counters.

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).

© Boardworks Ltd 2005

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.

Describe the locus made by the counters.

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.

© Boardworks Ltd 2005

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.

The locus is not rectangular, but is rounded at the corners.

Describe the locus made by the counters.

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).

© Boardworks Ltd 2005

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).

865.unknown

S9.4 Loci

* of 37

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.

© Boardworks Ltd 2005

* of 37

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.

867.unknown

The intersection of two loci

Suppose we have a red counter and a blue counter that are 9 cm apart.

9 cm

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.

Draw an arc of radius 6 cm from the blue counter.

Draw an arc of radius 5 cm from the red counter.

6 cm

6 cm

5 cm

5 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.

Before you begin make sure you have the following equipment:

A protractor

A pair of compasses

A sharp pencil

Constructions should be drawn in pencil and construction lines should not be rubbed out.

© Boardworks Ltd 2005

or

The lengths of all three sides (SSS)

The length of two sides and the included angle (SAS)

A right angle, the length of the hypotenuse and the length of one other side (RHS)

© Boardworks Ltd 2005

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.

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

© Boardworks Ltd 2005

1813.unknown

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

angle

angle

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.

© Boardworks Ltd 2005

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.

1814.unknown

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 pair of compasses.

Discuss how this triangle could be constructed using a ruler and compasses.

© Boardworks Ltd 2005

1815.unknown

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

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.

857.unknown

S9.5 Combining loci

S9.1 Constructing triangles

* of 37

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.

State that lines AB and CD are perpendicular.

© Boardworks Ltd 2005

* of 37

Bisecting angles

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

D

A

B

C

© Boardworks Ltd 2005

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.

1816.unknown

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.

1817.unknown

1818.unknown

1819.unknown

A

A

A

A

A

* of 37

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 traced by a football as it is kicked into the air and returns to the ground.

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

For example,

Ask pupils to describe the path of the ball by sketching it or tracing the path in the air with their fingers.

© Boardworks Ltd 2005

* of 37

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?

This shape is called a parabola.

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.

© Boardworks Ltd 2005

* of 37

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.

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?

Ask pupils to trace out the path using their finger tips or by sketching it. This path is similar the a sine curve.

© Boardworks Ltd 2005

* of 37

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?

How could you represent the path in two dimensions?

What about in three dimensions?

Ask pupils to trace out the path using their finger tips or by sketching it.

© Boardworks Ltd 2005

* of 37

Imagining regions

Franco promises free delivery for all pizzas within 3 miles of his Pizza House.

Can you describe the shape of the region within which Franco can deliver his pizzas free-of-charge?

3 miles

© Boardworks Ltd 2005

* of 37

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.

863.unknown

© Boardworks Ltd 2005

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

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).

© Boardworks Ltd 2005

Imagine placing counters so that their centres are always

3 cm from a line segment AB.

A

B

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.

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).

© Boardworks Ltd 2005

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

Describe the locus made by the counters.

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).

© Boardworks Ltd 2005

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.

Describe the locus made by the counters.

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.

© Boardworks Ltd 2005

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.

The locus is not rectangular, but is rounded at the corners.

Describe the locus made by the counters.

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).

© Boardworks Ltd 2005

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).

865.unknown

S9.4 Loci

* of 37

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.

© Boardworks Ltd 2005

* of 37

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.

867.unknown

The intersection of two loci

Suppose we have a red counter and a blue counter that are 9 cm apart.

9 cm

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.

Draw an arc of radius 6 cm from the blue counter.

Draw an arc of radius 5 cm from the red counter.

6 cm

6 cm

5 cm

5 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.