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KS4 Mathematics

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KS4 Mathematics

S9 Construction and loci

Contents

S9.2 Geometrical constructions

S9.1 Constructing triangles

S9.3 Imagining paths and regions

S9.4 Loci

S9.5 Combining loci

Equipment needed for constructions

A ruler marked in cm and mm

A protractor

A sharp pencil

A pair of compasses

Before you begin make sure you have the following equipment:

Constructing triangles

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

The size of two angles and a side (ASA)

The lengths of three of the sides (SSS)

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

To accurately construct a triangle you need to know:

Or

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.

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.

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 compass.

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

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.

Contents

S9.1 Constructing triangles

S9.2 Geometrical constructions

S9.3 Imagining paths and regions

S9.4 Loci

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

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

For example, in this diagram line BD bisects ABC.

A

B

C

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

D

Contents

S9.1 Constructing triangles

S9.2 Geometrical constructions

S9.3 Imagining paths and regions

S9.4 Loci

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.

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.

Imagining paths

A fluffy dice hangs from the rear-view mirror in a car and swings from side to side as the car moves forwards.

Can you imagine the path traced out by the dice?

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?

How could you represent the path in two dimensions?

What about in three dimensions?

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

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

Contents

S9.1 Constructing triangles

S9.2 Geometrical constructions

S9.4 Loci

S9.3 Imagining paths and regions

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

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 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.

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

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.

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.

Describe the locus made by the counters.

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

Contents

S9.1 Constructing triangles

S9.2 Geometrical constructions

S9.5 Combining loci

S9.3 Imagining paths and regions

S9.4 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 eat grass that is within 14 m of post R.

Describe how we could find the area that both goats can graze.

The intersection of two loci

6 cm

5 cm

6 cm

5 cm

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

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

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

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.