Download S9 Construction and loci

KS4 Mathematics
S9 Construction and loci
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Contents
S9 Construction and loci
S9.1 Constructing triangles
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:
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A ruler marked in
cm and mm
A protractor
A pair of compasses
A sharp pencil
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Constructing triangles
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 three of the sides (SSS)
Or
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
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.
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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?
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.
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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
side
side
Hint: We would need to use a compass.
We use the abbreviation SSS to stand for Side, Side, Side.
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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.
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.
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Constructing a triangle given RHS
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Contents
S9 Construction and loci
S9.1 Constructing triangles
S9.2 Geometrical constructions
S9.3 Imagining paths and regions
S9.4 Loci
S9.5 Combining 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.
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.
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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
D
B
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C
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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
S9 Construction and loci
S9.1 Constructing triangles
S9.2 Geometrical constructions
S9.3 Imagining paths and regions
S9.4 Loci
S9.5 Combining loci
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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.
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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.
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?
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Grazing sheep
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Contents
S9 Construction and loci
S9.1 Constructing triangles
S9.2 Geometrical constructions
S9.3 Imagining paths and regions
S9.4 Loci
S9.5 Combining loci
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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.
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.
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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
Q
Describe the locus made by the counters.
The locus is the perpendicular bisector of the line joining the
two points.
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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.
Describe the locus made by the counters.
The locus is the angle bisector of the angle where the two lines
intersect.
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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.
Describe the locus made by the counters.
The locus is not rectangular, but is rounded at the corners.
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The locus of points from a given shape
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Contents
S9 Construction and loci
S9.1 Constructing triangles
S9.2 Geometrical constructions
S9.3 Imagining paths and regions
S9.4 Loci
S9.5 Combining 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.
6 cm
Draw an arc of
radius 6 cm
from the blue
counter.
5 cm
9 cm
6 cm
5 cm
Draw an arc of
radius 5 cm
from the red
counter.
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.
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