Geometry

Written on 06 July 2020.

Reflections, Rotations and Enlargements

Sections:

0, Chapter 1, Introduction,
Introduction

After completing this unit you should
- understand how to apply the concept of reflection of a shape in a mirror line
- understand the concept of the rotation of a shape with a given angle, direction and centre
- be confident in making and identifying enlargements of 2-D shapes with given scale factors (and centres of enlargement).
This Unit describes how shapes can be transformed by reflection, rotation and enlargement.
You have three sections to work through and there are check up audits and fitness tests for each section.
1. Reflections
2. Rotations
3. Enlargements
0, Chapter 2, Reflections,
Reflections

Reflections are obtained when you draw the image that would be obtained in a mirror.

Every point on a reflected image is always the same distance from the mirror line as the original.

This is shown on the bottom.

Note:

Distances are always measured at right angles to the mirror line.
1, Chapter 2, Task 1, Worked Example 1,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Draw the reflection of the shape in the mirror line shown.
- Worked Example
  
  Reflect this shape in the mirror line shown in the diagram.
1, Chapter 2, Task 2, Exercise 1,
Exercise

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Copy the diagrams below and draw the reflection of each object.

Exercise 2

Copy the diagrams below and draw the reflection of each object.
0, Chapter 3, Rotations,
Rotations

Remember

Rotations are obtained when a shape is rotated about a fixed point, called the centre of rotation, through a specified angle.

The diagram shows a number of rotations.

It is often helpful to use tracing paper to find the position of a shape after a rotation.
1, Chapter 3, Task 1, Worked Example 2,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Rotate the triangle \(ABC\) shown in the diagram through \(90^\circ\) clockwise about the point with coordinates \(\left(0, 0\right)\).
- Worked Example
  
  The diagram shows the position of a shape \(A\) and the shapes, \(B, C, D, E\) and \(F\) which are obtained from \(A\) by rotation.
  
  Describe the rotation which moves \(A\) onto each other shape.
1, Chapter 3, Task 2, Exercise 2,
Exercise

Here are some questions to check your progress; there are more practice questions if needed.

1. Copy the axes and shape shown below.
1. Rotate the original shape through 90\(^\circ\) clockwise around the point (1, 2).
2. Rotate the original shape through 180\(^\circ\) around the point (3, 4).
3. Rotate the original shape through 90\(^\circ\) clockwise around the point (1, − 2).
4. Rotate the original shape through 90\(^\circ\) anti-clockwise around the point (0, 1).
2. The diagram shows the position of a shape labelled \(A\) and other shapes which were obtained by rotating \(A\).
1. Describe how each shape can be obtained from \(A\) by a rotation.
2. Which shapes can be obtained by rotating the shape \(E\)?
0, Chapter 4, Enlargements,
Enlargements

Remember

An enlargement is a transformation which enlarges (or reduces) the size of an image.
Each enlargement is described in terms of a centre of enlargement and a scale factor.

The example shows how the original, \(A\), was enlarged with scale factors \(2\) and \(4\).

A line from the centre of enlargement passes through the corresponding vertex of each image.

Remember

The distances, \(OA′\) and \(OA′′\), are related to \(OA\):
\(OA′ = 2 \times OA\)
\(OA′′ = 4 \times OA\)
The same is true of all the other distances between \(O\) and corresponding points on the images.
1, Chapter 4, Task 1, Worked Example 3,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Enlarge the triangle shown using the centre of enlargement marked and scale factor \(3\).
- Worked Example
  
  Enlarge the pentagon with scale factor \(2\) using the centre of enlargement marked on the diagram.
- Worked Example
  
  The diagram shows the square \(ABCD\) which has been enlarged to give the squares \(A′B′C′D′\) and \(A′′B′′C′′D′′\) .
  
  Find the centre of enlargement.
  
  Find the scale factor for each enlargement.
- Worked Example
  
  The diagram shows three triangles. \(ABC\) was enlarged with different scale factors to give \(A′B′C′\) and \(A′B′C′\).
  
  Find the centre of enlargement.
  
  Find the scale factor for each enlargement.
- Worked Example
  
  Enlarge the triangle shown with scale factor and centre of \(\frac{1}{4}\) enlargement as shown.
1, Chapter 4, Task 2, Exercise 3,
Exercise

Here are some questions to check your progress; there are more practice questions if needed

Exercise 1

Enlarge the shape with scale factor \(2\) using the point marked as the centre of enlargement.

Exercise 2

Enlarge the triangle shown with scale factor \(3\) and the centre of enlargement shown.

Exercise 3

For the diagram find the centre of enlargement and the scale factor when the smaller shape
is enlarged to give the bigger shape.

Exercise 4

For the diagram find the centre of enlargement and the scale factor when the smaller shape is enlarged to give the bigger shape.

Exercise 5

In each example below, the smaller shape has been obtained from the larger shape by an enlargement.
For each example, state the scale factor and the coordinates of the centre of enlargement.
0, Chapter 5, Summary,
Summary

Reflections

are obtained when you draw the image of a shape in a mirror line.
An example is shown below.

Rotations

are obtained when a shape is rotated about a point,
the centre of rotation, through a specified angle.
For example,

Transformation

moving a shape so that it is in a different position but retains the same size, area,
angles and line lengths.

An enlargement

is similar to a transformation but it alters (enlarges or reduces) the size of the
image. An enlargement is described in terms of a scale factor and also a centre of enlargement
which defines the exact location of the image.
For example, the image on the left below has been enlarged by a scale factor of \(2\frac{1}{2}\), with aspecified centre of enlargement.

The enlargement opposite (actually a reduction) of scale factor \(\frac{1}{3}\) has no specified centre
of enlargement, so the actual location of the smaller shape could be anywhere.

Interactive Exercises:

Reflections, Rotations and Enlargements Interactive Exercises, https://www.cimt.org.uk/sif/geometry/g7/interactive.htm
Reflections, https://www.cimt.org.uk/sif/geometry/g7/interactive/s1.html
Rotations, https://www.cimt.org.uk/sif/geometry/g7/interactive/s2.html
Enlargements, https://www.cimt.org.uk/sif/geometry/g7/interactive/s3.html

YouTube URL: https://youtu.be/JR_Ervr2654

File Attachments: media/course-resources/Reflections-Rotations-Enlargments_Essential-Informations.pdfmedia/course-resources/Reflections-Rotations-Enlargments_Learning-Objectives.pdfmedia/course-resources/Reflections-Rotations-Enlargments_Text.pdfmedia/course-resources/Reflection-Rotation-Enlargements_PowerPoint-Presentation.pptxmedia/course-resources/Reflections-Rotations-Enlargments_Answers.pdf

Written on 07 August 2020.

Congruence and Similarity

Sections:

0, Chapter 1, Introduction,
Introduction

This unit is focused on congruent and similar shapes and you need to understand the differences between these two concepts. After completing this unit you should
- understand the concept of congruent shapes and be able to use the congruence tests for triangles
- understand the concept of similar shapes and be able to determine similar shapes
You have two sections to work through namely;
1. Congruence
2. Similarity
0, Chapter 2, Congruence and Similarity,
Congruence and Similarity

Two shapes are said to be congruent if they are the same shape and size:
that is, the corresponding sides of both shapes are the same length and corresponding angles are the same.

The two triangles shown here are congruent.
1, Chapter 2.1, Similarity 1,
Similarity

Shapes which are of different sizes but which have the same shape are said to be similar.
The two triangles shown here are similar.

If you double the lengths for the triangle on the left, its sides will match those of the triangle on the right.
1, Chapter 2.2, Congruence Tests for Triangles,
Congruence Tests for Triangles

There are four tests for congruence with triangles.

TEST 1 (Side, Side, Side)

If all three sides of one triangle are the same as the lengths of the sides of the second triangle, then the two triangles are congruent.
This test is referred to as \(SSS\).

The single (double or treble) crosses on sides correspond to equal sides

TEST 2 (Side, Angle, Side)

If two sides of one triangle are the same length as two sides of the other triangle and the angle between these two sides is the same in both triangles, then the triangles are congruent.
This test is referred to as \(SAS\)

TEST 3 (Angle, Angle, Side)

If two angles and the length of one corresponding side are the same in both triangles, then they are congruent.
This test is referred to as \(AAS\).

The single or double arcs on angles correspond to equal angles

TEST 4 (Right angle, Hypotenuse, Side)

If both triangles contain a right angle, have hypotenuses of the same length and one other side of the same length, then they are congruent.
This test is referred to as \(RHS\).
1, Chapter 2.2, Task 1, Worked Example 1,
Worked Example
- Worked Example
  
  Which of the triangles below are congruent to the triangle \(ABC\), and why?
0, Chapter 3, Similarity,
Similarity

Similar shapes have the same shape but may be different sizes.
The two rectangles shown below are similar - they have the same shape
but one is smaller than the other.

They are similar because they are both rectangles and the sides of the larger rectangle are three times longer than the sides of the smaller rectangle.
1, Chapter 3.1, Similarity of Triangles,
Similarity of Triangles

These two triangles are not similar. The sides lengths of the triangles are not in the same ratio and so the triangles are not similar.

For two triangles to be similar, they must have the same internal angles, as shown in the similar shapes below.
1, Chapter 3.2, Similarity and Area,
Similarity and Area

It is interesting to compare the area of two similar rectangles. The area of the smaller rectangle is \(6\;cm^{2}\) and the area of the larger rectangle is \(54\;cm^{2}\), which is nine times \(\left(32\right)\) greater.

Remember

In general, if the lengths of the sides of a shape are increased by a factor \(k\), then the area is increased by a factor \(k^{2}\).
1, Chapter 3.3, Similarity and Volume,
Similarity and Volume

The diagrams below show 3 similar cubes.

Comparing the larger cubes with the \(1\;cm\) cube we can note that:

For the \(2\;cm\) cube
The lengths are \(2\) times greater.
The areas are \(4 \left(2^{2}\right)\) times greater.
The volume is \(8 \left(2^{3}\right)\) times greater.

For the \(3\;cm\) cube
The lengths are \(3\) times greater.
The areas are \(9 \left(3^{2}\right)\) times greater.
The volume is \(27 \left(3^{3}\right)\) times greater.

Remember

If the lengths of a solid are increased by a factor, \(k\), its surface area will increase by a factor \(k^{2}\) and its volume will increase by a factor \(k^{3}\).
1, Chapter 3, Task 1, Worked Example 2,
Worked Example
Use the slider to explore worked examples.
- Worked Example
  
  Which of the triangles, \(A, B, C, D\), shown below are similar?
  
  How do the areas of the triangles which are similar compare?
- Worked Example
  
  Explain why triangles \(ABE\) and \(ACD\) are similar.
  
  Find the lengths of \(x\) and \(y\).
  
  Find the ratio of the area of triangle \(ABE\) to triangle \(ACD\).
- Worked Example
  
  The diagrams show two similar triangles.
  
  If the area of triangle \(DEF\) is \(26.46\;cm^{2}\), find the lengths of its sides.
- Worked Example
  
  A can has a height of \(10\;cm\) and has a volume of \(200\;cm^{3}\). A can with a similar shape has a height of \(12\;cm\).
  
  Find the volume of the larger can.
  
  Find the height of a similar can with a volume of \(675\;cm^{3}\).
0, Chapter 4, Summary,
Summary

Congruent shapes

Congruent shapes are identical but can have different orientations.
For example, these five shapes are all congruent.

Similar shapes

Similar shapes have corresponding angles equal and corresponding sides in the same ratio.
For example, shapes \(A, B, C, D\) and \(E\) are all similar; \(B\) and \(D\) are also congruent.

Tests for congruency

There are four tests for congruency of triangles, namely
- Side, Side, Side (\(SSS\))
- Angle, Angle, Side (\(AAS\))
- Side, Angle, Side (\(SAS\))
- Right angle, Hypotenuse, Side (\(RHS\))

Interactive Exercises:

Congruence and Similarity Interactive Exercises, https://www.cimt.org.uk/sif/geometry/g4/interactive.htm
Congruence, https://www.cimt.org.uk/sif/geometry/g4/interactive/s1.html
Similarity, https://www.cimt.org.uk/sif/geometry/g4/interactive/s2.html

YouTube URL: https://youtu.be/iRUrwVxMuD8

File Attachments: media/course-resources/Congruence-Similarity_Essential-Informations.pdfmedia/course-resources/Congruence-Similarity_Answers.pdfmedia/course-resources/Congruence-Similarity_Text.pdfmedia/course-resources/Congruence-Similarity_PowerPoint-Presentation.pptxmedia/course-resources/Congruence-Similarity_Learning-Objectives.pdf

Written on 10 August 2020.

Coordinates

Sections:

0, Chapter 1, Introduction,
Introduction

This is the first of three units based on analytic geometry, which brings algebra and geometry together. The other two units are Straight Lines and Using Graphs to Solve Equations. After completing this unit you should
- be able to identify and illustrate points in two dimensions with positive coordinates
- be able to identify and illustrate points in two dimensions
- be confident in plotting straight lines
- be confident in plotting curves
- be able to find the mid-point of a line segment.
Coordinates are used to describe position in two dimensions. This Unit will show you how to use coordinates and apply them to some problems.

You have five sections to work through namely;
1. Positive Coordinates
2. Coordinates
3. Plotting Straight Lines
4. Plotting Curves
5. Mid-Points of Line Segments
0, Chapter 2, Positive Coordinates,
Positive Coordinates

Remember

Coordinates are pairs of numbers that uniquely describe a position on a rectangular grid.

These numbers are sometimes referred to as Cartesian coordinates.

The first number refers to the horizontal \(\left(x-axis\right)\) and the second the vertical \(\left(y-axis\right)\). Note that axes should always be labelled.
1, Chapter 2, Task 1, Worked Example 1,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Plot the points with coordinates \(\left(3, 8\right)\), \(\left(6, 1\right)\) and \(\left(2, 5\right)\).
- Worked Example
  
  Write down the coordinates of each point in the diagram below.
  
  Find the maximum volume of water the extinguisher could hold.
0, Chapter 3, Coordinates,
Coordinates

Remember

The coordinates of a point are written as a pair of numbers, \(\left(x, y\right)\), which describe where the point is on a set of axes. The axes can include positive and negative values for \(x\) and \(y\).

The \(x-axis\) is always horizontal (i.e. across the page) and the \(y-axis\) always vertical (i.e. up the page).

The \(x-coordinate\) is always given first and the \(y-coordinate\) second.
1, Chapter 3, Task 1, Worked Example 2,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  On a grid, plot the point A which has coordinates \(\left(–2, 4\right)\), the point \(B\) with coordinates \(\left(3, –2\right)\) and the point \(C\) with coordinates \(\left(– 4, –3\right)\).
- Worked Example
  
  Write down the coordinates of each place on the map of the island.
0, Chapter 4, Plotting Straight Lines,
Plotting Straight Lines

Use the slider to explore worked examples.

By calculating values of coordinates you can find points and draw a graph for any relationship, such as \(y = 2x - 5\). This is best demonstrated by studying the following worked examples.
- Worked Example
  
  Copy and complete the following pairs of coordinates using the relationship \(y = x − 2\).
  \(\left(4, ?\right), \left(1, ?\right), \left(–1, ?\right)\)
  
  Plot the points on a set of axes and draw a straight line through them.
- Worked Example
  
  Draw the graph of \(y = 2x − 1\).
0, Chapter 5, Plotting Curves,
Plotting Curves

Some relationships produce curves rather than straight lines when plotted. This is shown in the worked examples below

Use the slider to explore worked examples.
- Worked Example
  
  a. Complete the table below using the relationship \(y = x^{2} - 2\).
  
  b. Write a list of coordinates using the data in the table.
  
  c. Plot the points and draw a smooth curve through them.
- Worked Example
  
  Draw the graph of \(y = x^{3} − 4x\) for values of \(x\) from \(–3\) to \(3\).
0, Chapter 6, Mid-Points of Line Segments,
Mid-Points of Line Segments

The coordinates of the mid-point between two other points may be found by drawing or by calculation
The mid Point of a line segment is the point which is exactly in the middle of it.

The diagram below shows the mid-point of the coordinates \(\left(2, 2\right)\) and \(\left(6, 8\right)\); the coordinate of the mid-point is \(\left(4, 5\right)\).

You can see from the diagram that the \(x-coordinate\) is in fact the mean value of the two \(x-coordinates\) of the end points of the line segment. Similarly the \(y-coordinate\) is the mean value of the two \(y-coordinates\) of the end points of the line segment.

Remember

Given any two points with coordinates \(\left(a, b\right)\) and \(\left(c, d\right)\), the coordinates of the mid-point of the line segment joining the two points is given by;

\(\left(\frac{a+b}{2},\frac{c+d}{2}\right)\)
1, Chapter 6, Task 1, Worked Example 3,
Worked Example
Worked Example
Find the coordinates of the mid-point of the line segment:

\(AB\)

\(AC\)

\(BD\)
0, Chapter 7, Summary,
Summary

Coordinates

\(\left(x, y\right)\) means that the point \(P\) with coordinates \(\left(a, b\right)\) is such that \(x = a\), \(y = b\)

Coordinate axes

These are shown on the diagram below.

Straight line

Is defined by any two points on the line \(P\) and \(Q\) in the diagram below.

Curves

These are lines that are not straight (see diagram below).

Line segment

This is any part of a straight line between two points.

Mid-point of line segment

This is the point \(X\) in the line segment \(PQ\), such that length \(PX = length\;XQ\).
The coordinates of \(X\) are:
\(\left(\frac{a+c}{2},\frac{b+d}{2} \right)\)

Interactive Exercises:

Coordinates Interactive Exercises, https://www.cimt.org.uk/sif/geometry/g5/interactive.htm
Positive Coordinates, https://www.cimt.org.uk/sif/geometry/g5/interactive/s1.html
Coordinates, https://www.cimt.org.uk/sif/geometry/g5/interactive/s2.html
Plotting Straight Lines , https://www.cimt.org.uk/sif/geometry/g5/interactive/s3.html
Plotting Curves, https://www.cimt.org.uk/sif/geometry/g5/interactive/s4.html
Mid-Points of Line Segments, https://www.cimt.org.uk/sif/geometry/g5/interactive/s5.html

YouTube URL: https://youtu.be/oU5cFukpRdI

File Attachments: media/course-resources/Coordinates_Learning-Objectives.pdfmedia/course-resources/Coordinates_PowerPoint-Presentation.pptxmedia/course-resources/Coordinates_Text.pdfmedia/course-resources/Coordinates_Answers.pdfmedia/course-resources/Coordinates_Essential-Informations.pdf

Written on 11 August 2020.

Straight Lines

Sections:

0, Chapter 1, Introduction,
Introduction

This is the second unit in this strand with the first being Coordinates. Here we focus on the straight line. After completing this unit you should:
- understand the concept of gradient
- be able to calculate the gradient of a straight line
- understand how to use straight line graphs in practical problems such as velocity/time graphs
- be able to calculate the equation of a straight line
You have four sections to work through namely;
1. Gradient
2. Gradients of Perpendicular Lines
3. Applications of Graphs
4. Equation of a Straight Line
0, Chapter 2, Gradient,
Gradient

Remember

The gradient of a line describes how steep it is.

The diagram below shows two lines, one with a positive gradient and the other with a negative gradient.

Top Tip

Think of a gradient as going up or down a hill.
A positive gradient can be seen to go ‘uphill’ in moving from left to right and a negative gradient goes ‘downhill’
1, Chapter 2.1, Calculating Gradient,
Calculating Gradient

Remember

The gradient of a line between two points, \(A\) and \(B\), is calculated using the following formula
\(gradient\;of\;AB=\frac{vertical\;change}{horizontal\;change}\)

For a line as shown in the diagram below the gradient is calculated as

\(\frac{\left(y - coordinate\;of\;B\right) -\left(y -coordinate\;of\;A\right) }{\left(x - coordinate\;of\;B\right) -\left(x -coordinate\;of\;A\right)} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Note that parallel lines have the same gradient.
1, Chapter 2.1, Task 1, Worked Example 1,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Find the gradient of the line shown in the diagram.
- Worked Example
  
  Find the gradient of the line joining the point \(A\) with coordinates \(\left(2, 4\right)\) and the point \(B\) with coordinates \(\left(4, 10\right)\).
- Worked Example
  
  Find the gradient of the line that joins the points with coordinates \(\left(–2, 4\right)\) and \(\left(4, 1\right)\).
- Worked Example
  
  The coordinates of the points \(A,\;B,\;C\) and \(D\) are listed below.
  
  \(A\;\left(2, 4\right)\) \(B\;\left(8, 7\right)\) \(C\;\left(–1, –5\right)\) \(D\;\left(5, –2\right)\)
  
  Show that the line segments \(AB\) and \(CD\) are parallel.
  
  Are the line segments \(AC\) and \(BD\) parallel?
0, Chapter 3, Gradients of Perpendicular Lines,
Gradients of Perpendicular Lines

Remember

The product of the gradients of two perpendicular lines will always be \(–1\), unless one of the lines is horizontal and the other is vertical.

This was found in the previous example with the perpendicular lines of \(AB\) and \(PQ\):
the gradient of \(AB = 3\): the gradient of \(PQ = -\frac{1}{3}\) and \(3\times-\frac{1}{3}=-1\)

An equivalent way of describing this is by showing
\(gradient\;of\;AC= \frac{-1}{gradient\;AB}\)

If the gradient of a line is \(m\), and \(m \neq 0\),then the gradient of a perpendicular line will be \(-\frac{1}{m}\)
1, Chapter 3, Task 1, Worked Example 2,
Worked Example

Use the slider to explore worked examples.
In this section we explore the relationship between the gradients of perpendicular (at right angles to each other) lines and line segments.
- Worked Example
  
  Plot the points \(A \left(1, 2\right)\) and \(B \left(4, 11\right)\), join them to form the line \(AB\) and then calculate the gradient of \(AB\).
  
  On the same set of axes, plot the points \(P \left(5, 4\right)\) and \(Q \left(8, 3\right)\), join them to form the line \(PQ\) and calculate the gradient of \(PQ\).
  
  Measure the angle between the lines \(AB\) and \(PQ\). What do you notice about the two gradients?
- Worked Example
  
  Show that the line segment joining the points \(A\;\left(3, 2\right)\) and \(B\;\left(5, 7\right)\) is perpendicular to the line segment joining the points \(P\;\left(2, 5\right)\) and \(Q\;\left(7, 3\right)\).
0, Chapter 4, Conversion Graphs,
Conversion Graphs

In this section some applications of graphs are considered, particularly conversion graphs and graphs to describe motion.

The graph below is a conversion graph: it can be used for converting US Dollars into and from Jamaican Dollars.

Note ,as currency exchange rates are continually changing, these rates might not be correct now.
1, Chapter 4.1, Distance-Time Graphs,
Distance-Time Graphs

The following graph is used to describe motion.

A distance-time graph of a car is shown below. The gradient of this graph gives the speed of the car. The gradient is steepest from \(A\) to \(B\), so this is when the car has the greatest speed. The gradient \(BC\) is zero, so the car is not moving.

Note that we sometimes use speed and other times velocity. In fact, velocity is the speed in a given direction.
1, Chapter 4.2, Velocity/Speed-Time Graphs,
Velocity/Speed-Time Graphs

The following graph is also used to describe motion.

The area under a speed/velocity-time graph gives the distance travelled. Finding the shaded area on the graph shown opposite would give the distance travelled.

The gradient of this graph gives the acceleration of the car. There is constant acceleration from \(0\) to \(20\) seconds, then zero acceleration from \(20\) to \(40\) seconds (when the car has constant speed), constant deceleration from \(40\) to \(50\) seconds, etc.
1, Chapter 4.2, Task 1, Worked Example 3,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  A temperature of \(20^{\circ}\) is equivalent to \(68^{\circ} F\) and a temperature of \(100^{\circ}C\) is equivalent to a temperature of \(212 ^{\circ}F\). Use this information to draw a conversion graph. Use the graph to convert:
  
  \(30^{\circ}\) to \(^{\circ}Farenheit\)
  
  \(180^{\circ}\) to \(^{\circ}Celsius\)
- Worked Example
  
  The graph shows the distance travelled by a girl on a bike.
  
  Find the speed she is travelling on each stage of the journey.
  Note that units are m/s (metres per second), as m are the units for distance and s the units for time.
- Worked Example
  
  The graph shows how the velocity of a bird varies as it flies between two trees. How far apart are the two trees?
- Worked Example
  
  The velocity-time graph below, not drawn to scale, shows that a train stops at two stations,
  \(A\) and \(D\). The train accelerates uniformly from \(A\) to \(B\), maintains a constant speed
  from \(B\) to \(C\) and decelerates uniformly from \(C\) to \(D\).
  
  Using the information on the graph,
  
  Calculate, in \(ms^{−2}\), the train's acceleration
  
  Show that the train took \(30\;secs\) from \(C\) to \(D\) if it decelerated at \(\frac{1}{2}ms^{−2}\) .
  
  If the time taken from \(A\) to \(D\) is \(156\;seconds\), calculate the distance in metres between the two stations.
0, Chapter 5, Equation of a Straight Line,
Equation of a Straight Line

Remember

The equation of a straight line is usually written in the form \(y=mx+c\) where \(m\) is the gradient and \(c\) is the \(y\) intercept.

Note - the intercept is the point where the line crosses the \(y\;axis\). See the diagram below.
1, Chapter 5, Task 1, Worked Example 4,
Worked Example

Use the slider to explore worked examples.
In this section we explore the relationship between the gradients of perpendicular (at right angles to each other) lines and line segments.
- Worked Example
  
  Find the equation of the line shown in the diagram.
- Worked Example
  
  Find the equation of the line that passes through the points \(\left(1, 5\right)\) and \(\left(3, 1\right)\).
- Worked Example
  
  Draw the line \(𝑥 = 4\).
  
  Draw the line \(𝑦 = 2\).
  
  Write down the coordinates of the point of intersection of these lines.
- Worked Example
  
  The point with coordinates \(\left(4, 9\right)\) lies on the line with equation \(y=2x+1\). Determine the equation of the perpendicular line that also passes through this point.
- Worked Example
  
  In the diagram below, not drawn to scale, AB is the straight line joining \(A \left(−1, 9\right)\) and \(B \left(3, 1\right)\).
  
  Calculate the gradient of the line, \(AB\).
  
  Determine the equation of the line, \(AB\).
  
  Write the coordinates of \(G\), the point of intersection of \(AB\) and the \(y-axis\)
  
  Write the equation of the line through \(O\), the origin, that is perpendicular to \(AB\).
  
  Write the equation of the line through \(O\) that is parallel to \(AB\).
0, Chapter 6, Summary,
Summary

Gradient of a straight line

The gradient describes how steep the line is and is defined as

\(m = \frac{vertical\;change}{horizontal\;change}\)

\(=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\)

The gradient of a line can be positive or negative.

Parallel lines

Lines with the same gradient are parallel to one another.

Perpendicular lines

The product of their gradients is \(−1\) ; for example, see diagram.

In general, if \(m\) is the gradient of one of the lines \(\left(-\frac{1}{m}\right)\) is the gradient of the other line.

Distance-time graph

Here the gradient represents the velocity;

Velocity-time graph

Here the gradient represents the acceleration; the area under the graph is the distance travelled.

General equation of a straight line

The equation is \(y = mx + c\) when \(m\) is the gradient and \(c\) is the \(y-axis\) intercept.

Interactive Exercises:

Straight Lines Interactive Exercises, https://www.cimt.org.uk/sif/geometry/g8/interactive.htm
Gradient, https://www.cimt.org.uk/sif/geometry/g8/interactive/s1.html
Gradients of Perpendicular Lines, https://www.cimt.org.uk/sif/geometry/g8/interactive/s2.html
Applications of Graphs, https://www.cimt.org.uk/sif/geometry/g8/interactive/s3.html
The Equation of a Straight Line, https://www.cimt.org.uk/sif/geometry/g8/interactive/s4.html

YouTube URL: https://youtu.be/8-VbcaXbZpo

File Attachments: media/course-resources/Straight-Lines_PowerPoint-Presentation.pptxmedia/course-resources/StraightLines_Text.pdfmedia/course-resources/Straight-Lines_Answers.pdfmedia/course-resources/Straight-Lines_Essential-Informations.pdfmedia/course-resources/Straight-Lines_Learning-Objectives.pdf

Written on 12 August 2020.

Bearings

Sections:

0, Chapter 1, Introduction,
Introduction

This unit continues the angle geometry theme, with the important topic of bearings. After completing
this unit you should
- be able to identify and use bearings in practical problems.
0, Chapter 2, Bearings,
Bearings

A compass bearing tells us direction.

The 4 main directions are North, South, East and West.
We can remember the order (starting from north, and going clockwise) as ‘Never Eat Shredded Wheat’:

Note that the angle between \(N\) and \(E\) is \(90^{\circ}\) and between \(N\) and \(NE\) is \(45^{\circ}\)

Notice that the bearings in between start with either North or South:
North-East, South-East, South-West, North-West.
1, Chapter 2.1, In between directions,
In between directions

In between these bearings we have:
- North-North-East (\(NNE\))*
- East-North-East (\(ENE\))
- East-South-East (\(ESE\))
- South-South-East (\(SSE\))
- South-South-West (\(SSW\))
- West-South-West (\(WSW\))
- West-North-West (\(WNW\))
- North-North-West (\(NNW\))
Notice the bearings in between start with:
North, East, South or West:

NNE, ENE, ESE, SSE, SSW, WSW, WNW, NNW

* NNE can be read as North of North-East
1, Chapter 2.2, Calculating a Bearing,
Calculating a Bearing

Three figure bearings can also be used as an alternative to compass bearings.

They are measured in a particular way:
Remember
How to calculate a bearing
- We start measuring from North
- Measure the angle in degrees \(^{\circ}\) clockwise
- The bearing (angle) is given as three figures;often this means we may have to ‘add’ zeros at the start. For example: North-East is 045 \(^{\circ}\)
The advantage of 3-figure bearings is that they can describe direction uniquely.

Note: Sometimes 3-figure bearings are given with such accuracy that they have digits after a decimal point,
e.g. 192.75\(^{\circ}\) (this is still called a 3-figure bearing, as it has 3 figures (digits) before the decimal point).
1, Chapter 2.3, 3-Figure Bearings,
3-Figure Bearings

Examples of 3-figure bearings:

Bearings can be especially useful in navigation, be it at sea or for people walking in the wild on open moorland or hills.
1, Chapter 2.3, Task 1, Worked Example,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Work out the bearing of \(B\) from \(A\).
- Worked Example
  
  a) Write down the bearing of A from P.
  
  b) b) Work out the bearing of B from P.
- Worked Example
  
  The bearing of a ship (S) from a lighthouse (L) is \(050^{\circ}\) Work out the bearing of the lighthouse from the ship.
0, Chapter 3, Summary,
Summary

Bearings

A bearing gives the direction or position of something, or the direction of movement, relative to a fixed point.
Bearings are of the form of angles, expressed in degrees as three-digit numbers; they are measured from north
in a clockwise direction.

In finding bearings, say \(A\) from \(B\), you must find the angle (measured clockwise) made by the \(N\) direction with the line from \(B\) to \(A\).
For example, \(060^{\circ}\), \(210^{\circ}\).

Interactive Exercises:

Compass Bearings, https://www.cimt.org.uk/sif/geometry/g3/interactive/s1.html

YouTube URL: https://youtu.be/8-6xmZWwsYs

File Attachments: media/course-resources/Bearings_Learning-Objectives.pdfmedia/course-resources/Bearings_PowerPoint-Presentation.pptxmedia/course-resources/Bearings_Text.pdfmedia/course-resources/Bearings_Answer.pdfmedia/course-resources/Bearings_Essential-Information.pdf

Written on 12 August 2020.

Angles and Symmetry

Sections:

0, Chapter 1, Introduction,
Introduction

This is the first of two units on angle geometry; the second is Angles, Circles and
Tangents. Angle geometry underpins all the work on trigonometry in the set of units that
follows. After completing this unit you should
- be able to measure and construct accurately, using a protractor, angles up to \(360^{\circ}\)
- be able to identify the rotational symmetry and line symmetry of \(2\) dimensional shapes
- know and understand the angle facts for points, lines, triangles and quadrilaterals
- know and understand the angle facts for parallel and intersecting lines
- be able to determine the size of the interior and exterior angles for a regular polygon.
You have five sections to work through namely;
1. Measuring Angles
2. Line and Rotational Symmetry
3. Angle Geometry
4. Angles with Parallel and Intersecting Lines
5. Angle Symmetry in Regular Polygons
0, Chapter 2, Measuring Angles,
Measuring Angles

Remember

An angle is a measure of turn. Angles can be measured in degrees:
\(180\) degrees is a half turn, \(360\) degrees a full turn and \(90\) degrees a quarter turn, known as a right angle.

Degrees are indicated with a small circle, e.g. \(90^{\circ}\) represents \(90\) degrees.

The angle around a complete circle is \(360^{\circ}\), a full turn.

The angle around a point on a straight line is \(180^{\circ}\), a half turn.

A protractor can be used to measure or draw angles which we show in the next example.
1, Chapter 2, Task 1, Worked Examples 1,
Worked Examples

Use the slider to explore worked examples.
- Worked Example
  
  Measure the angle \(CAB\) in the triangle shown.
- Worked Example
  
  Measure the marked angle.
- Worked Example
  
  Draw angles of
  
  \(120^{\circ}\)
  
  \(330^{\circ}\)
0, Chapter 3, Rotational Symmetry,
Rotational Symmetry

An object has rotational symmetry if it can be rotated about a point so that it fits on top of itself without completing a full turn.

The shapes below have rotational symmetry.

In a complete turn this shape fits on top of itself two times.
It has rotational symmetry of order \(2\).

In a complete turn this shape fits on top of itself four times.
It has rotational symmetry of order \(4\).
0, Chapter 4, Lines of Symmetry,
Lines of Symmetry

Shapes have line symmetry if a mirror could be placed so that one side is an exact reflection of the other.
These imaginary 'mirror lines' are shown by dotted lines in the diagrams below.

This shape has \(2\) lines of symmetry.

This shape has \(4\) lines of symmetry.
1, Chapter 4, Task 1, Worked Example 2,
Worked Examples
- Worked Example
  
  For the given shape, state:
  
  the number of lines of symmetry,
  
  the order of rotational symmetry.

0, Chapter 5, Angle Geometry,

Angle Geometry

There are a number of important results concerning angles in different shapes, at a point and on a line.

There are 6 rules that you must remember :

Before we describe the rules let us remind you of the names for different angles


Name	Definition
ACUTE angle	Less than \(90\) degrees
OBTUSE angle	Between \(90\) degrees and \(180\) degrees
RIGHT angle	\(90\) degrees
REFLEX angle	Between \(180\) degrees and \(360\) degrees

1, Chapter 5.1, Rules at a point or on a line,
Rules at a point or on a line

Angles at a Point
The angles at a point will always add up to \(360^{\circ}\).
It does not matter how many angles are formed at the point - their total will always be \(360^{\circ}\).

Angles on a Line
Any angles that form a straight line add up to \(180^{\circ}\).
1, Chapter 5.2, Rules in triangles and quadrilaterals,
Rules in triangles and quadrilaterals

Angles in a Triangle
The angles in any triangle add up to \(180^{\circ}\).

Angles in an Equilateral Triangle

In an equilateral triangle all the angles are \(60^{\circ}\) and all the sides are the same length.

Note that the tick marks mean lines of equal length

Angles in an Isosceles Triangle

In an isosceles triangle two sides are the same length and two angles are the same size.

Angles in a Quadrilateral

The angles in any quadrilateral add up to \(360^{\circ}\).
1, Chapter 5.2, Task 1 , Worked Example 3,
Worked Examples

Use the slider to explore worked examples.
- Worked Example
  
  Find the sizes of angles \(𝑎\) and \(𝑏\) in the diagram on the right.
- Worked Example
  
  Find the angles \(𝑎\), \(b\), \(c\) and \(d\) in the diagram.
- Worked Example
  
  In the figure, not drawn to scale, \(ABC\) is an isosceles triangle with
  
  \(\angle CAB = p^{\circ}\) and \(\angle ABC = \left(p+3\right)^{\circ}\).
  
  Write an expression in terms of \(p\) for the value of the angle at \(C\)
  
  Determine the size of each angle in the triangle
0, Chapter 6, Angles with Parallel and Intersecting Lines,
Angles with Parallel and Intersecting Lines

Opposite Angles

When any two lines intersect, two pairs of equal angles are formed.
The two angles marked \(a\) are a pair of opposite equal angles.
The angles marked \(b\) are also a pair of opposite equal angles.

Corresponding Angles

When a line intersects a pair of parallel lines, \(angle\;a= angle\;b\); the \(angles\;a\) and \(b\) are called corresponding angles.

Alternate Angles

The angles \(c\) and \(d\) are equal, and are called alternating angles

Supplementary Angles

The angles \(𝑏\) and \(c\) add up to \(180^{\circ}\), and are called supplementary angles.
1, Chapter 6, Task 1, Worked Example 4,
Worked Examples

Use the slider to explore worked examples.
- Worked Example
  
  Find the sizes of angles \(𝑎\) and \(𝑏\) and \(c\) in the diagram on the right.
- Worked Example
  
  Find the sizes of angles \(𝑎\) and \(𝑏, c\) and \(d\) in the diagram on the right.
- Worked Example
  
  Find the angles \(𝑎\) and \(𝑏, c\) and \(d\) in the diagram.
- Worked Example
  
  In the diagram on the right, not drawn to scale, \(AB\) is parallel to \(CD\) and \(EG\) is parallel to \(FH\), angle \(IJL =50^{\circ}\) and angle \(KIJ =95^{\circ}\)
  
  Calculate the values of \(x, y\) and \(z,\) showing clearly the steps in your calculations.
0, Chapter 7, Angle Symmetry in Regular Polygons,
Angle Symmetry in Regular Polygons

Remember

Regular polygons have sides which are all equal in length; all interior angles are equal too.
Regular polygons will have both line and rotational symmetry.

This symmetry can be used to find the interior angles of a regular polygon.
1, Chapter 7, Task 1, Worked Examples 5,
Worked Examples

Use the slider to explore worked examples.
- Worked Example
  
  Find the interior angle of a regular dodecagon (\(12\) sides).
- Worked Example
  
  Find the sum of the interior angles of a regular heptagon.
- Worked Example
  
  Copy the octagon shown in the diagram and draw in any lines of symmetry.
  
  Copy the octagon and shade in extra triangles so that it now has rotational symmetry.
0, Chapter 8, Summary,
Summary

An object has rotational symmetry if it can be rotated about a point so that it fits on top of itself without completing a full turn.
The shapes below have rotational symmetry.

In a complete turn this shape fits on top of itself two times.
It has rotational symmetry of order \(2\).

In a complete turn this shape fits on top of itself four times.
It has rotational symmetry of order \(4\).

Shapes have line symmetry if a mirror could be placed so that one side is an exact reflection of the other. These imaginary 'mirror lines' are shown by dotted lines in the diagrams below.

This shape has \(2\) lines of symmetry

This shape has \(4\) lines of symmetry.

Angles at a point

The angles at a point will always add up to \(360^{\circ}\). It does not matter how many angles are formed at the point – their total will always be \(360^{\circ}\).

Angles on a line

Adjacent angles that form a straight line add up to \(180^{\circ}\)

Angles in a triangle

The angles in any triangle add up to \(180^{\circ}\)

Angles in an equilateral triangle

In an equilateral triangle all the angles are \(60^{\circ}\) and all the sides are the same length.

Angles in an isosceles triangle

In an isosceles triangle two sides are the same length and the two base angles are the same size.

Angles in a quadrilateral

The angles in any quadrilateral add up to \(360^{\circ}\) .

Opposite angles

When any two lines intersect, two pairs of equal opposite angles are formed. The two angles marked \(a\) and \(c\) are a pair of opposite equal angles, so angle \(a\) is equal to angle \(c\).
The angles marked \(b\) and \(d\) are also a pair of opposite equal angles, so angle \(b\) is equal to angle \(d\)

Corresponding angles

When a line intersects a pair of parallel lines, angle \(a\) is equal to angle \(b\). The angles \(a\) and \(b\) are called corresponding angles.

Alternate angles

The angles \(c\) and \(d\) are equal.

Interactive Exercises:

Interactive Exercises - Angles and Symmetry, https://www.cimt.org.uk/sif/geometry/g1/interactive.htm
Measuring Angles, https://www.cimt.org.uk/sif/geometry/g1/interactive/s1.html
Line and Rotational Symmetry, https://www.cimt.org.uk/sif/geometry/g1/interactive/s2.html
Angle Geometry, https://www.cimt.org.uk/sif/geometry/g1/interactive/s3.html
Angles with Parallel and Intersecting Lines, https://www.cimt.org.uk/sif/geometry/g1/interactive/s4.html
Angle Symmetry in Regular Polygons, https://www.cimt.org.uk/sif/geometry/g1/interactive/s5.html

YouTube URL: https://youtu.be/--rxHv3GL9g

External Links:

Angles and triangles, https://www.bbc.co.uk/bitesize/guides/zrck7ty/revision/4, images/Images/bitesize-logo.png, The four types of angle you should know are acute, obtuse, reflex and right angles. When you are estimating the size of an angle, you should consider what type of angle it is first.

File Attachments: media/course-resources/Angles-Symmetry_Essential-Information.pdfmedia/course-resources/Angles-Symmetry_Learning-Objectives.pdfmedia/course-resources/Angles-Symmetry_PowerPoint-Presentation.pptxmedia/course-resources/Angles-Symmetry_Text.pdfmedia/course-resources/Angles-Symmetry_Answers.pdf

Written on 13 August 2020.

Graphs

Sections:

0, Chapter 1, Introduction,
Introduction

A graph is a diagram showing the relationship between some variable quantities.

This unit uses the concept of graphical functions in order to solve equations.

After completing this unit you should
- be able to graph linear simultaneous equations in order to find the solution
- understand the graphical form of well-known functions; these include linear, quadratic, cubic
  and reciprocal functions
- be able to graph functions in order to solve equations, including the solution of quadratic and linear equations.
You have four sections to work through namely;
1. Solution of Simultaneous Equations by Graphs
2. Graphs of Common Functions
3. Graphical Solutions of Equations
4. Tangents to Curves
0, Chapter 2, Simultaneous Equations,
Simultaneous Equations - Introduction

Try solving: \(x + y = 24\)
What could ‘\(x\)’ be? What could ‘\(y\)’ be?

There are simply too many possible solutions.

So if we have more than one ‘unknown’ (variable), we must have more than one equation.

With two ‘unknowns’, we must have at least two equations, and we must solve them at the same time, i.e. simultaneously.

Let us try again:

\(x+y=24\) \((1)\)

\(2x=20\) \((2)\)

Can you solve it now?

What could \(x\) be?
What could \(y\) be?

First, label the equations as shown.

From the equation 2, we can work out \(x=10\)

So now we can use this solution and ‘plug’ it in (i.e. substitute) this value into the equation 1.

\(x+y=24\)
Substitute \(x\) value into equation 1 \(10+y=24\)
Now solve for \(y\): \(y=14\)

We can see therefore that the solutions are:
\(x=10\)
\(y=14\)
1, Chapter 2.1, Solving by Graphs,
Solving by Graphs

There are three ways to solve simultaneous linear equations
- One method is by substitution which we have jst seen in the last section
- Another method is by elimination
- In this unit we will look at solving simultaneous linear equations by drawing graphs (and finding the coordinates of where the 2 lines cross)
Now lets show the graphical method in an example by find solutions for \(x\) and \(y\) that satifies these two equations \(x + y = 24\)
\(2x = 20\)

First we rearrange the equations and write in the form \(y = …\)

From the first equation,
\(x + y = 24\)
\(y = 24 - x\)

From the second equation,
\(2x=20\)
\(x=10\)

These are both linear (straight line) equations.
To draw a straight line, only two pairs of coordinates are required.

For: \(y = 24 - x\),
if \(x = 0\), then \(y = 24\) and
if \(x = 24\), then \(y = 0\)

We can see that \(\left(0,24\right)\) and \(\left(24,0\right)\) are both points on the line

For \(x=10\)
for each and every \('y'\) coordinate \(x=10\)

This is the vertical line going through \(x=10\)

We can now draw the graphs defined by our two equations and idenitify the point where they cross as shown below.
1, Chapter 2.1, Task 1, Worked Example 1,
Worked Example
- Worked Example
  
  By drwawing graphs find solutions for \(x\) and \(y\) that satifies these two equations?
  \(x + y = 8\)
  \(2x + 3y = 21\)
  
  First we rearrange the equations and write in the form \(y = …\)
  
  From the first equation,
  \(x + y = 8\)
  \(y = 8 - x\)
  
  From the second equation,
  \(\begin{align*} 2x+3y&=21\\ 3y&=21-2x\\ y&=21-2x\\ y&=\frac{21-2x}{3}\\ y&=7-\frac{2x}{3} \end{align*}\)
  These are both linear (straight line) equations.
  To draw a straight line, only two pairs of coordinates are required.
  
  For: \(y = 8 - x\),
  if \(x = 0\), then \(y = 8\) and
  if \(x = 8\), then \(y = 0\)
  \(\left(0,8\right)\) \(\left(8,0\right)\) are on the line
  
  For: \(y=7-\frac{2x}{3}\)
  if \(x=0\), then \(y=7\) and
  if \(x=3\), then \(y=5\)
  \(\left(0,7\right)\) \(\left(3,5\right)\) are on the line
  
  These points are then plotted:
  \(\left(0,8\right)\) \(\left(8,0\right)\)
  and
  \(\left(0,7\right)\) \(\left(3,5\right)\)
  
  The two lines intersect at the point \(\left(3,5\right)\), so the solution is:
  \(x=3\) and \(y=5\)
  
  Note - Graphing software (such as DESMOS) can be used to plot and explore graphs effortlessly.
0, Chapter 3, Graphs of Common Functions,
Graphs of Common Functions

Remember

Linear Functions is the name given to equations whose graph is a straight line. Such equations as always of the form \(y +mx +c\)

The graph is shown below. \(c\) is the point where the graphs cross the \(y\;axis\) and is called the intercept.
1, Chapter 3.1, Quadratic Functions,
Quadratic Functions

Remember

Quadratic functions contain an \(x^{2}\) term as well as multiples of \(x\) and a constant.

The following graphs show three examples:
1, Chapter 3.2, Cubic Functions,
Cubic Functions

Remember

Cubic functions contain an \(x^{3}\) term as well as multiples of \(x^{2}, x\) and a constant.

Here are some examples below
1, Chapter 3.3, Reciprocal ufnctions,
Reciprocal Functions

Remember

Reciprocal functions have the form of a fraction with \(x\) as the denominator, for example \(y =\frac {5}{2x}\).

The following graphs show three examples:
1, Chapter 3.4, Expotential Functions,
Expotential Functions

Remember

Exponential functions contain a term with the variable (unknown) as the power (or index or exponent),
for example \(y =2^{x}\).

The following graph shows the graph of \(y =2^{x}\).
1, Chapter 3.4, Task 1, Worked Example 2,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Write down the letter of the graph which could have the equation:
  
  a. \(y = 3x - 2\)
  
  b. \(y = 2x^{2} + 5x - 3\)
  
  c. \(y = \frac {3}{x}\)
- Worked Example
  
  This sketch below shows part of the graph with equation \(y = pq^{x}\) where \(p\) and \(q\) are constants. The points with coordinates \(\left(0, 8\right)\), \(\left(1, 18\right)\) and \(\left(1.5, k\right)\) lie on the graph.
  
  Work out the values of \(p\), \(q\) and \(k\).
0, Chapter 4, Graphical Solutions of Equations,
Graphical Solutions of Equations

Lets explore solving more complex equations

In fact we can solve more complex equations in exactly the same way

We draw a graph of both equations and find the points where the graphs cross. This give us the solutions.

Ofcourse it is possible that the graphs may not cross in which case the orginal equations do not have any common solutions

This is best demonstated by some examples, which are shown in the next section
1, Chapter 4, Task 1, Worked Example 3,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  How can we solve: \(x^{2}=x+\frac{1}{x}\) by a graphical method?
- Worked Example
  
  How would you find the solution to:
  \(2x - x^{2} = x^{3}\) ?
0, Chapter 5, Tangents to Curves,
Tangents to Curves

A tangent is a line that touches a curve at one point only, as shown here.

The gradient of the tangent gives the gradient of the curve at that point. A gradient of the curve gives the rate at which a quantity is changing.

For example, the gradient of a distance-time curve gives the rate of change of distance with respect to time, which actually gives us the velocity.

You may have seen tangents whilst working with circles.

Here is an example of two tangents to points on a circle.

\(BA\) and \(BC\) each touch the circle at just one point.
1, Chapter 5, Task 1, Example 1,
Example

Below we have drawn the graph of \(y = x^{3}\) for \(-2 \leq x \leq 2\).

On the graph we have drawn the tangents to the curve at \(x = -1\) and \(x = 1\).

We can find the gradients of the tangents at these points from the graph.

By drawing the triangles as shown under each tangent, we can see that the gradients of both tangents are \(3\).
0, Chapter 6, Summary,
Summary

Simultaneous (linear) equations

can be solved by finding the point of intersection of the two straight lines.

They take the form

\(ax+by=c\)
\(dx+ey=f\)
(\(a,b,c,d,e\) and \(f\) constants)

Quadratic functions

are parabolic in shape. For example,

There are of the form \(f\left(x\right) = ax^{2} + bx + c\) \(\left(a \neq 0\right)\)
For example, \(f\left(x\right) = 2x^{2} - x+1\)

Quadratic equations

are of the form \(ax^{2} + bx + c = 0\)
\(\left(a \neq 0\right)\)
For example, \(2x^{2} - x + 1= 0\)

Cubic functions

They are of the form \(f\left(x\right) = ax^{3} + bx^{2} + cx + d\) (\(a \neq 0\))

For example, \(f\left(x\right) =x^{3}\), \(f\left(x\right) =2x^{3}-x+1\)

Cubic equations

are of the form \(ax^{3} +bx^{2}+cx+d=0\)
\(\left(a \neq 0\right)\)

For example, \(2x^{3}-x+1=0\)

Reciprocal functions

are of the form

They are the form \(f\left(x\right) = \frac{k}{x}\)

For example, \(f\left(x\right)=\frac{1}{x}, f\left(x\right)=-\frac{2}{x}\)

Roots of a quadratic

equation there can be \(2\) or \(1\) or \(0\) roots for a quadratic equation.
For example,

Roots of a cubic equation

there can be 3 or 2 or 1 root of a quadratic equation.
For example,

Tangent

a line that touches a curve at one pointonly, as shown opposite.

Interactive Exercises:

Graphs Interactive Exercises, https://www.cimt.org.uk/sif/geometry/g6/interactive.htm
Solution of Simultaneous Equations by Graphs, https://www.cimt.org.uk/sif/geometry/g6/interactive/s1.html
Graphs of Common Functions, https://www.cimt.org.uk/sif/geometry/g6/interactive/s2.html
Graphical Solutions of Equations, https://www.cimt.org.uk/sif/geometry/g6/interactive/s3.html
Tangents to Curves, https://www.cimt.org.uk/sif/geometry/g6/interactive/s4.html

YouTube URL: https://youtu.be/UOki-4Zvx_w

File Attachments: media/course-resources/Graphs_Learning-Objectives.pdfmedia/course-resources/Graphs_PowerPoint-Presentation.pptxmedia/course-resources/Graphs_Text.pdfmedia/course-resources/Graphs_Answers.pdfmedia/course-resources/Graphs_Essential-Informations.pdf

Written on 15 September 2020.

Angles, Circles and Tangents

Sections:

0, Chapter 1, Introduction,
Introduction

This unit continues the angle geometry theme, covering
circles and tangents. After completing this unit you should
- understand and be able to use angle geometry results for circles, chords, diameters and tangents.
You have three sections to work through; Results 1- 4 are given in Section 1, Results 5-7 in Section 2 and Results 8-10 in Section 3.
1. Angles and Circles 1
2. Angles and Circles 2
3. Circles and Tangents
0, Chapter 2, Angles and Circles 1,
Angles and Circles 1

The following result is true in any circle.

Result 1

When a triangle is drawn in a semi-circle as shown, the angle on the circumference is always a right angle.

Proof

Join the centre, \(O\), to the point, \(P\), on the perimeter as shown in the diagram

Notice now that the lines \(OA\), \(OP\) and \(OB\) are all equal, as each is a radius of the circle.

Since \(OB = OP\) (equal radii) then \(angle\;OBP = angle\;OPB\) ( \(= x\), as labelled) because triangle \(OBP\) is isosceles.

Similarly, triangle \(AOP\) is also isosceles, so \(angle\;OAP = angle\;APO\) ( \(= 𝑦\), say)

In triangle \(ABP\), the sum of the angles must be \(180 ^{\circ}\) so ,

\(\begin{align*} 𝑦 + \left(𝑦 + x\right) + x&=180 ^{\circ}\\ 2 x + 2𝑦&=180 ^{\circ}\\ x + 𝑦&=90 ^{\circ} \end{align*}\)

But \(angle\;APB = x + 𝑦\), so this is a right angle, as the result states.
1, Chapter 2.1, Tangent,
Tangent

Remember

A tangent is a line that touches only one point on the circumference of a circle.

This point is known as the point of tangency.

Result 2

A tangent is always perpendicular to the radius of the circle.
1, Chapter 2.2, Chord,
Chord

Remember

A chord is a line joining any two points on the circle.

The perpendicular bisector is a second line that divides the chord in half and is at right angles to it.

See both shown in the diagram below

Result 3

The perpendicular bisector of a chord always passes through the centre of the circle.

Result 4

When the ends of a chord are joined to the centre of a circle, an isosceles triangle is formed, so the two angles marked are equal.
1, Chapter 2.2, Task 1, Worked Example 1,
Worked Examples

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- Worked Example
  
  Find the angles marked with letters in the diagram, if \(O\) is the centre of the circle.
- Worked Example
  
  Find the angles \(a\), \(b\) and \(c\), if \(AB\) is a tangent and \(O\) is the centre of the circle.
- Worked Example
  
  Find the angles marked in the diagram, where \(O\) is the centre of the circle.
0, Chapter 3, Angles and Circles 2,
Angles and Circles 2

The following further results are true in any circle.

Result 5

The angle subtended by an arc, \(PQ\), at the centre is twice the angle subtended at the circumference.

Proof

\(OP = OC\) (equal radii), therefore \(angle\;CPO = angle\;PCO\)

Similarly, \(angle\;CQO = angle\;QCO\)

We label the angles as \(x\) and \(y\), as shown in the diagram above

Now, extending the line CO to D, say and looking at the triangle \(POC\) we see that,

\(\begin{align*} angle\;POD&=x + x\\ &=2x \end{align*}\)

Similarly,

\(\begin{align*} angle\;QOD&=𝑦 + 𝑦\\ &=2𝑦 \end{align*}\)

Hence,

\(\begin{align*} angle\;POQ&=2x+2𝑦\\ &=2\left(x+𝑦\right)\\ &=2\times angle\;PCQ \end{align*}\)
1, Chapter 3.1, Subtended Angles,
Subtended Angles

Result 6

Angles subtended at the circumference by a chord (on the same side of the chord) are equal; that is, in the diagram

\(a = b\)

Proof

Because angle \(𝑎\) is on the circumference, the angle at the centre is \(2𝑎\), according to Result 5.
But if you consider the angle \(𝑏\), which is also at the circumference, the angle at the centre is \(2𝑏\), according to Result 5.

Thus \(2𝑎 = 2𝑏\); divide both sides to obtain \(𝑎 = 𝑏\), as required.
1, Chapter 3.2, Cyclic quadrilaterals,
Cyclic quadrilaterals

In cyclic quadrilaterals (quadrilaterals where all \(4\) vertices lie on a circle), opposite angles sum to \(180^{\circ}\):

\(a + c = 180^{\circ}\)

and

\(b + d = 180^{\circ}\)

Proof

Construct the diagonals \(AC\) and \(BD\), as shown.

Then label the angles subtended by \(AB\) as w; that is \(angle\;ADB = angle\;ACB\) (= w)

Similarly for the other chords, the angles being marked \(x, 𝑦\) and \(z\) as shown.

Now that we have labelled the angles, as shown on the right, we can prove the result

In triangle \(ABD\), the sum of the angles is \(180^{\circ}\) , so

\(w + z + \left(x + 𝑦\right) = 180^{\circ}\)

You can rearrange this as

\(\left(x + w\right) + \left(𝑦 + z\right) = 180 ^{\circ}\)

which shows that angle \(CDA + angle\;CBA = 180^{\circ}\)

This proves that the angle at \(D\) and the angle at \(B\), the opposite sides on the quadrilateral, add to \(180^{\circ}\).

The result that the angles at \(A\) and \(C\) add to \(180 ^{\circ}\) can be shown in a similar way.
1, Chapter 3.2, Task 1, Worked Example 2,
Worked Example

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  Find the angles marked with letters in the diagram, if \(O\) is the centre of the circle.
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- Worked Example
  
  Find the angles marked in the diagram. \(O\) is the centre of the circle
- Worked Example
  
  In the diagram the line \(AB\) is a diameter and \(O\) is the centre of the circle. Find the angles marked.
- Worked Example
  
  In the diagram the chords \(AB\) and \(CD\) are parallel. Prove that the triangles \(ABE\) and \(DEC\) are isosceles.
0, Chapter 4, Circles and Tangents,
Circles and Tangents

Three further important results are now presented.

Result 8

If two tangents are drawn from a point \(T\) to a circle with centre \(O\), and \(P\) and \(R\) are the points of contact of the tangents with the circle, then, using symmetry,
- \(PT = RT\)
- Triangles \(TPO\) and \(TRO\) are congruent*
* Important note - Congruent shapes are identical; that is, corresponding angles are equal and corresponding sides are equal
1, Chapter 4.1, Alternate Segment,
Alternate Segment

Result 9

The angle between a tangent and a chord equals an angle at the circumference subtended by the same chord;

e.g. \(𝑎 = 𝑏\) in the diagram.

This is known as the alternate segment theorem and needs a proof, as it is not obviously true!

Proof

We need to show that \(angle\;RPT = angle\;PQR\)

Construct the diameter \(POS\), as shown.

We know that angle \(SRP = 90^{\circ}\) since \(PS\) is a diameter.

Now, \(angle\;PSR = angle\;PQR = x\), say and

\(\begin{align*} angle\;SPR&= 180^{\circ} − 90 ^{\circ} − x\\ &= 90^{\circ} − x \end{align*}\)

But

\(\begin{align*} angle\;RPT&= 90^{\circ} − \left(angle\;SPR\right)\\ &=90 ^{\circ} − \left(90 ^{\circ} − x ^{\circ}\right)\\ &=x\\ &=angle\;PQR \end{align*}\)
1, Chapter 4.2, Intersecting chords,
Intersecting chords

Result 10

For any two intersecting chords, as shown,

\(AX \times CX = BX \times DX\)

Proof

In triangles \(AXB\) and \(DXC\), \(angle\;BAC = angle\;BDC\)

(equal angles subtended by chord \(BC\))

and \(angle\;ABD = angle\;ACD\) (equal angles subtended by chord \(AD\))

This tells us that \(AXB\) and \(DXC\) are similar*,

It follows that \(\frac {AX}{BX} = \frac{DX}{CX}\)

And \(AX \times CX = BX \times DX\) as required.

* Important note - Similar shapes differ only in size so that corresponding angles are the same and corresponding lengths are in the same ratio
1, Chapter 4.3, Special Case,
Special Case

Note that these follow from particular cases of Result 10:

This result will still be true even when the chords intersect outside the circle, as illustrated below

When the chord \(BD\) becomes a tangent, and \(B\) and \(D\) coincide at the point P

\(AX \times CX = PX \times PX\)

\(AX \times CX = PX^{2}\)
1, Chapter 4.3, Task 1, Worked Example 3,
Worked Example
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- Worked Example
  
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- Worked Example
  
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0, Chapter 5, Summary,
Summary

A bearing gives the direction or position of something, or the direction of movement, relative to a fixed point.
Bearings are of the form of angles, expressed in degrees as three-digit numbers; they are measured from north in a clockwise direction.

For example, \(060^{\circ}\;210^{\circ}\)

A tangent is a line that touches only one point on the circumference of a circle.
A tangent is always perpendicular to the radius of the circle.

The point of tangency is the point where a tangent touches the circumference of a circle.

A chord is a line joining any two points on the circle.
The perpendicular bisector is a second line that divides the first line in half and is at right angles to it.
The perpendicular bisector of a chord always passes through the centre of the circle.

An angle subtended by a chord in a circle is shown in the diagram opposite. Angles subtended at the circumference on the same side of a circle by a chord are equal.

An angle subtended at the centre of a circle by an arc is twice the angle subtended at a point on the circumference (alternate segment theorem).

Interactive Exercises:

Angles, Circles and Tangents Interactive Exercises, https://www.cimt.org.uk/sif/geometry/g2/interactive.htm
Angles and Circles 1, https://www.cimt.org.uk/sif/geometry/g2/interactive/s1.html
Angles and Circles 2, https://www.cimt.org.uk/sif/geometry/g2/interactive/s2.html
Circles and Tangents, https://www.cimt.org.uk/sif/geometry/g2/interactive/s3.html

YouTube URL: https://youtu.be/gexkjriVEs8

File Attachments: media/course-resources/Angles-Circles-Tangents_Essential-Informations.pdfmedia/course-resources/Angles-Circles-Tangents_Learning-Objectives.pdfmedia/course-resources/Angles-Circles-Tangents_PowerPoint-Presentation.pptxmedia/course-resources/Angles-Circles-Tangents_Text.pdfmedia/course-resources/Angles-Circles-Tangents_Answers.pdf

Reflections, Rotations and Enlargements

Introduction

Reflections

Worked Example

Worked Example

Worked Example

Exercise

Rotations

Remember

Worked Example

Worked Example

Worked Example

Exercise

Enlargements

Remember

Remember

Worked Example

Worked Example

Worked Example

Worked Example

Worked Example

Worked Example

Exercise

Summary

Reflections

Rotations

Transformation

An enlargement

Congruence and Similarity

Introduction

Congruence and Similarity

Similarity

Congruence Tests for Triangles

Worked Example

Worked Example

Similarity

Similarity of Triangles

Similarity and Area

Remember

Similarity and Volume

Remember

Worked Example

Worked Example

Worked Example

Worked Example

Worked Example

Summary

Congruent shapes

Similar shapes

Tests for congruency

Coordinates

Introduction

Positive Coordinates

Remember

Worked Example

Worked Example

Worked Example

Coordinates

Remember

Worked Example

Worked Example

Worked Example

Plotting Straight Lines

Worked Example

Worked Example

Plotting Curves

Worked Example

Worked Example

Mid-Points of Line Segments

Remember

Worked Example

Worked Example

Summary

Coordinates

Coordinate axes

Straight line

Curves

Line segment

Mid-point of line segment

Straight Lines