GCSE Mathematics – Geometry and Algebra

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Contents

1. Proportionality 3

2. Map Scales 7

3. Trial and improvement 9

4. Quadratic Formula 12

5. Sketching Graphs 19

6. Transformations 29

7. Area and Perimeter 36

8. Points, Lines and Angles and Triangles 40

9. Circles 47

10. Enlargements 49

11. Trigonometry 54

12. Triangles inside circles 60

13. Constructions 66

14. Volume 73

15. Cylinders 75

16. Spheres and Cones 80

1 Proportionality

Proportionality describes the relationship between two variables where the relationship between the two variables can be expressed in terms of a proportionality constant.

Direct proportionality

The variable y is said to be directly proportional to the variable x if their relationship can be described as

y=cx

where c is a constant.

The above can be re-written as , i.e. the ratio y/x is constant.

Direct proportionality can be observed when plotting y against x.

As x increases y increases.

The rate of the increase is the proportionality constant, 3.

Inverse proportionality

The variable y is said to be inversely proportional to the variable x if their relationship can be described as

This can be written as yx = c, i.e. the product yx is constant.

If y is inversely proportional to x then y decreases as x increases.

Proportion to powers

Direct and inverse proportionality can also be to the powers of x.

For example we could write:

y is directly proportional to x

y is directly proportional to

y is directly proportional to

y is directly proportional to

and

y is inversely proportional to x

y is inversely ...

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