Semester 1 Summary
General Maths Summary Semester 1
Measurement
Pythagoras Theorem
For any right angled triangle
In any right triangle:
Pythagoras Theorem
The square on the hypotenuse = sum of the squares of the other
two sides.
𝑐 2 = 𝑎2 + 𝑏 2
Areas and Perimeters (Formula)
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Volume
Formula
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1 m3 = 1kL
Ratio and Proportion
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We need to convert to the same units.
Dividing a given quantity in a given ratio
Proportions
Example
A metal bar 42 cm long weighs 1.5 kg. Find the weight of a bar 72 cm long and of the same metal.
Method 1 (Unitary Method)
42 cm weighs 1.5 kg
1 cm weighs
1.5
42
= 0.035714𝑘𝑔
72 cm weighs 72 × 0.035714 = 2.5714 𝑘𝑔
Method 2 (Using proportions)
Let the unknown weight be 𝑥.
Form a proportion:
𝑥
72
=
1.5
42
𝑥=
1.5 × 72
= 2.5714 𝑘𝑔
42
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Semester 1 Summary
Network Summary
Notice that the network above has multiple edges joining nodes a to d and the network has a loop
at node d. Also c is called an isolated node as it is not connected to any other node.
Degree of a Node
Every node (or vertex) has a degree, which is the number of paths leaving the node. In the diagram
above node a has degree 3, node b has degree 2, node c has degree 0, and node d has degree 5.
Note that for node d the loop counts twice towards the degree. Nodes a and d have odd degrees
and nodes c and b have even degrees.
The sum of the degrees of any network = 𝟐 × number of edges in the network.
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Connected Network (or graph): If there is a path between any two vertices of a graph, then the
graph is connected.
The graph below is connected as all vertices are connected.
The graphs below are disconnected since there is no path connecting A to D.
Planar Graphs
A graph that can be drawn without intersecting edges is called a planar graph.
Note that at first sight a graph may appear not to be planar; the question is whether it can be
redrawn without intersections. If it can then the graph is planar.
The graph on the right at first sight appears non planar as
the edges intersect.
However, it can be redrawn as shown on the right, by
“stretching” one of the edges.
Both graphs are clearly planar.
Some graphs cannot be transformed into planar graphs,
such as the one on the right:
Euler’s Formula
A planar graph divides the plane into a number of regions (sometimes called faces). Remember that
the area outside the graph is also a region.
The graph shown has 6 regions, numbered 1 to 6.
There is an important connection between the number of vertices,
number of edges and number of regions.
For any connected planar graph:
V+R=E+2
Where V is the number of vertices, R is the number of regions (or
faces) and E is the number of edges.
For the graph shown V = 4, R = 6, E = 8
So 4 + 6 = 8 + 2
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Euler and Hamiltonian Paths
Euler Paths and Circuits
A network is traversible if it can be drawn or traced without lifting pencil from paper and without
retracing any edge. The network must be connected and a path must exist that passes over each
edge exactly once.
The above describes an Euler path. In an Euler path you trace over every edge exactly once.
An Euler circuit is an Euler path which starts and finishes at the same vertex.
1. Euler Path that is also an Euler Circuit
If all the vertices of the graph (network) have even degree then an Euler path exists. Also the path
forms an Euler Circuit. A circuit occurs when you start and finish at the same vertex (or node). It
does not matter which vertex you start with you will always finish at that same vertex. The following
are examples of Euler circuits. Note that you are allowed to pass through a vertex more than once.
2. Euler Path
An Euler path exists if the graph has exactly two vertices of odd degree. The path can only be traced
by starting at one of the vertices of odd degree and finishing at the other vertex of odd degree. You
cannot form a circuit.
The following are examples of Euler paths.
A
You must start at q and finish at
t or start at t and finish at q.
B
You must start at A and finish at
B or start at B and finish at A
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You must start at F and finish
at A or start at A and finish at
F.
Semester 1 Summary
The following graphs do not form Euler paths or Euler circuits as the above rules do not hold.
More than 2 vertices of odd
degree.
More than 2 vertices of odd
degree
More than 2 vertices of odd
degree
Euler paths are useful for deciding on the most efficient route. Think of a road repair gang or the
refuse engineers who empty the garbage bins. They would not want to reverse their trucks back the
way they came. This wastes time. They would be interested in a route in which they do not have to
back track. They would often prefer to start and end at the same point, traversing a circuit.
Hamiltonian Paths and Circuits
A Hamiltonian path uses every vertex exactly once. It is important to realise that not all edges need
to be used.
Hamiltonian Path
There are often several Hamiltonian paths in the same graph.
The path 1 – 2 – 3 – 4 – 5 forms a Hamiltonian
path as every vertex is visited once only. There
are other solutions.
Hamiltonian Circuit
A Hamiltonian circuit is a Hamiltonian path that starts and ends at the same vertex.
The path 1 – 2 – 3 – 5 – 4 – 1 forms a
Hamiltonian circuit as every vertex is visited once
only apart from the starting vertex.
There are other solutions.
There are no rules for finding Hamiltonian paths or circuits. You can trace them out by inspection.
Travelling sales people are often interested in Hamiltonian paths or circuits. They would not want to
revisit the same town or area twice.
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Trees
In any tree the number of vertices = number of edges + 1
Minimum Spanning Tree
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Rather than try to find the minimum spanning tree by trial and error we can use the following steps:
Step 1. Choose any vertex and select the smallest edge leading from it.
Step 2. Inspect the 2 vertices included so far and select the smallest edge leading from either
vertex. If there is a tie, arbitrarily choose any one.
Step 3. Inspect all vertices included so far and select the smallest edge leading from any included
vertex. If there is a tie, arbitrarily choose any one.
Step 4: Repeat step 3 until all vertices in the network are included in the tree.
Example 1
1. Choose the vertex A to start with.
Inspect the network and find the shortest edge
connected to A. This is edge AD.
2. Inspect A and D and find the shortest edge
connecting one of these to a third vertex. This is
DB
3. Inspect A, D and B and find the shortest edge
connecting one of these to another vertex. This
is BC.
Since all the vertices have been connected this is the minimum spanning tree.
The total length of this spanning tree is 9 + 10 + 13 = 32 km. Thus the minimum amount of cable
necessary is 32 km, arranged as shown in the last diagram.
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Example 2
Starting with node a look for the shortest edge connected to a. Repeat the steps to determine the
minimum spanning tree as shown below. Notice that there are 2 solutions.
Shortest Path
The shortest path between two nodes can often be found by inspection.
Example 1.
By inspection the shortest path between A and F is A-C-E-F = 5 + 8 + 6 = 19 m
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Example 2.
On inspection the shortest path between vertex 1 and vertex 9 is: 1-2-3-6-7-9
which is 4 + 3 + 3 + 2 + 2 = 14
Notice that the shortest path always forms a tree.
Linear Equations and Linear Graphs
Solving Simultaneous Equations
Method 1: Using graphs
Sketch 2 graphs and
find point of
intersection.
Method 2: using substitution
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Method 3 using elimination
Method 4 use calculator
Try solving the set of simultaneous equations
opposite using the calculator.
In Calculator View
Menu – Algebra – Solve System of Equations
– Solve System of Linear Equations
Enter the equations into the templates
The calculator method is often the best
method in application problems when the
numbers can be awkward.
2𝑥 =
20 24
+
5
5
2𝑥 =
44
5
Gives 𝑥 =
22
5
Simultaneous Equations and Applications
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Linear Equations
Gradient or Slope of a Straight Line
Remember that the gradient of a line gives a measure of the steepness of the line and its sign is
important.
Positive gradient
Negative gradient
Zero gradient
Gradient not defined
The general equation of line is
𝒚 = 𝒎𝒙 + 𝒄 , where m is the gradient and c is the y-intercept
Remember the special cases
𝒚 = 𝒎𝒙
always passes through the
origin
𝑥=4
is a vertical line passing
through 𝑥 = 4
𝑦 = −3
is a horizontal line passing
through 𝑦 = −3
Graph Sketching
When sketching straight line graphs, you only need to find two points through which the graph
passes. The easiest way is to find the intercepts.
Take 𝑥 = 0 and find the y-intercept
Take 𝑦 = 0 and find the x-intercept.
If the graph passes through the origin, you will need to test a different value of x to find the y value.
The coordinates can be plotted and joined with a ruler to form the linear graph.
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Linear Modelling
In application problems, linear graphs are usually drawn in the first quadrant. Using the equation of
the graph, you will need to be able to interpret the gradient in the context of the question and the yintercept.
You may also have to form a linear equation based on the variables given in the question.
Remember:
If the y value increases as the x value increases then the graph has a positive gradient
If the y value decreases as the x value increases then the graph has a negative gradient
Also, check out the videos in the Linear Graphs section of General Maths on STLLink
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