Further Mathematics

Geometry & Trigonometry Summary

Introduction

In this lesson we will consider how we can choose the right technique to use for a given problem.

This will include… 1.

Things to do when starting a question 2.

Choosing the right technique 3.

Things to check before you finish

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

2. Choosing the right approach

To get started we will divide all of the possible questions into five groups.

1.

Problems involving perimeters 2.

3.

4.

5.

Problems involving areas Problems involving volumes Problems involving similar figures Problems involving lengths and angles of triangles

2.1 Problems involving perimeter

 Find the total distance around the outside of the shape.

 For questions involving circles use C = 2πr

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 use Area = ½ ab sin C For problems involving Area, 3 sides use Heron’s Formula (see page 422)

2.3 Problems involving volume

 Prisms  Use V prism = Area of cross section  height  Pyramids & Cones  Use V pyramid = 1/3 Area of base  height  Spheres  Use V sphere = 4/3πr 3  Composite shapes  Divide the shape into prisms, pyramids & cones and spheres. Find the volume of each and add them to get the total.

Examples

 Find the perimeter of this shape.

 Find the area.

Examples

 Find the total surface area.

 Find the volume.

Examples

 Find the area.

 Find the area.

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 = k 2 and vsf = k 3

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 use obtuse angle = 180 ° - acute angle  For problems involving 3 sides, 1 angle use the Cosine rule.

 To find an unknown side:

a

2 

b

2 

c

2  2

bc

cos

A

 To find an unknown angle: cos

A



b

2 

c

2 

a

2 2

bc

Examples

 What is the angle at B?

 What is the angle s?

Examples

 What is the angle of elevation?

 What is the length of the unknown side?

3. Before you finish

 Don’t 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?