© boardworks ltd 2005 1 of 37 s9 construction and loci ks4 mathematics
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© Boardworks Ltd 2005 1 of 37
S9 Construction and loci
KS4 Mathematics
© Boardworks Ltd 2005 2 of 37
Contents
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AS9.1 Constructing triangles
S9 Construction and loci
S9.2 Geometrical constructions
S9.3 Imagining paths and regions
S9.4 Loci
S9.5 Combining loci
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Equipment needed for constructions
Before you begin make sure you have the following equipment:
A protractorA ruler marked in
cm and mm
A pair of compasses A sharp pencil
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Constructing triangles
To accurately construct a triangle you need to know:
or
To accurately construct a triangle you need to know:
The size of two angles and a side (ASA)
The lengths of three of the 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)
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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.
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Constructing a triangle given SAS
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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
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Constructing a triangle given ASA
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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.
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Constructing a triangle given SSS
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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
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Constructing a triangle given RHS
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S9.2 Geometrical constructions
Contents
S9.3 Imagining paths and regions
S9.5 Combining loci
S9.1 Constructing triangles
S9 Construction and loci
S9.4 Loci
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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.
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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
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The perpendicular bisector of a line
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The bisector of an angle
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The perpendicular from a point to a line
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The perpendicular from a point on a line
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Contents
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S9.3 Imagining paths and regions
S9.5 Combining loci
S9.2 Geometrical constructions
S9.1 Constructing triangles
S9 Construction and loci
S9.4 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 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,
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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.
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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?
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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?
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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
Franco’s Pizza House is not
drawn to scale!
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Grazing sheep
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Contents
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S9.4 Loci
S9.5 Combining loci
S9.2 Geometrical constructions
S9.1 Constructing triangles
S9 Construction and loci
S9.3 Imagining paths and regions
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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
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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.
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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.
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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.
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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.
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The locus of points from a given shape
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Contents
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S9.5 Combining loci
S9.2 Geometrical constructions
S9.1 Constructing triangles
S9 Construction and loci
S9.3 Imagining paths and regions
S9.4 Loci
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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.
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Tethered goats
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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.
6 cm
6 cm
5 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.