Further Mathematics

1
Further Mathematics

• Geometry Trigonometry
• Summary

2
Introduction

• In this lesson we will consider how we can choose
  the right technique to use for a given problem.
• This will include
• Things to do when starting a question
• Choosing the right technique
• Things to check before you finish

3
1.Starting a question

• Read the question carefully.
• Draw a diagram and list any values that have been
  given.
• Add any extra information that can be easily
  worked out using geometry laws
• Eg If you have two angles in a triangle find the
  third (180 other two angles).
• Convert from bearings to angles
• Double check the question for more information
• Eg for similar figures, which one is the original

4
2. Choosing the right approach

• To get started we will divide all of the possible
  questions into five groups.
• Problems involving perimeters
• Problems involving areas
• Problems involving volumes
• Problems involving similar figures
• Problems involving lengths and angles of triangles

5
2.1 Problems involving perimeter

• Find the total distance around the outside of the
  shape.
• For questions involving circles useC 2pr

6
2.2 Problems involving area

• Simple shapes
• Choose from the formulas on p332
• Composite shapes
• Divide the shape into simple shapes
• Total Surface Area of a 3D shape
• For common shapes choose from the formulas on
  p338
• For other shapes draw a net and add the areas of
  each face (p339)
• For triangles where base and height are not
  known
• For problems involving Area, 2 sides, 1 angle
  useArea ½ ab sin C
• For problems involving Area, 3 sides use Herons
  Formula (see page 422)

7
2.3 Problems involving volume

• Prisms
• Use Vprism Area of cross section ? height
• Pyramids Cones
• Use Vpyramid 1/3 Area of base ? height
• Spheres
• Use Vsphere 4/3pr3
• Composite shapes
• Divide the shape into prisms, pyramids cones
  and spheres. Find the volume of each and add them
  to get the total.

8
Examples

• Find the perimeter of this shape.
• Find the area.

9
Examples

• Find the total surface area.
• Find the volume.

10
Examples

• Find the area.
• Find the area.

11
2.4 Problems involving similar figures

• Proving similarity
• Use AAA, SSS (or for similar triangles SAS)
• Finding the scale factor
• Use k length on copy length on original
• Finding lengths using k
• Use the ratios of corresponding sides or
• Use the scale factor (above).
• Problems involving areas and volumes
• Use lsf k, asf k2 and vsf k3

12
2.5 Problems involving lengths and angles of
triangles

• Right angled triangles
• For problems involving 3 sides use Pythagoras
  theorem
• For problems involving 2 sides and 1 angle use
  Trigonometric ratios (SOHCAHTOA)
• Triangles that do not have a right angle
• For problems involving 2 sides, 2 angles use the
  Sine rule.
• To find an obtuse angle useobtuse angle 180 -
  acute angle
• For problems involving 3 sides, 1 angle use the
  Cosine rule.
• To find an unknown side
• To find an unknown angle

13
Examples

• What is the angle at B?
• What is the angle s?

14
Examples

• What is the angle of elevation?
• What is the length of the unknown side?

15
3. Before you finish

• Dont forget the last step in the calculation
• Did you need to take the square root?
• Did you need to use an inverse trig function
  (sin-1, cos-1 or tan-1)
• Have you shown the correct units?
• Have you used the right number of decimal places?
• If the answer was an angle
• Should it be converted to a bearing?
• Should it be in degrees and minutes?
• Have you answered the question?