# Maths

Module 5

## Straight line graphs

General equation -- y = mx + c

c = y intercept (where the line cuts the y axis)

m= y/x

If 2 lines cross at a right angle the gradients are the same but with different signs.

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## Vectors

A vector quantity has both magnitude and direction

Vectors can be used to represented physical quantities such as force, velocity and acceleration.

A displacement vector caan be display on a co-ordinate grid as a directed line segment.

Vectors are denated by a bold letter and a column pair.

2 vectors are equal if they have the same magnitude and direction.

The magnitude of a vector can be found using pythagoras' theorem.

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## Vectors

The magnitude of a vector can be calculated without being displayed on a grid.

a = (x over y) = square root of (x squared + y squared)

Vectors that are represented by line segments can be added using the nose to tail method.

To obtain the resultant vector a+b, the tail of b is joined to the nose of a.

Eg.. a+b = (3 over 3)+(4 over 1) = (7 over 4) = resultant vector of a+b

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## Translation Vector

Used to describe the movement of a shape around a grid to give a cungruent shape.

( no reflection or rotation has taken place )

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## Transformation of the Graph y = f(x)

The graph of -f(x) is a reflection of f(x) in the x axis.

The graph of kf(x) gives a stretch of f(x) by scale factor K in the y axis.

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## Inductory Terminology - circle theorems

Arc AB subtends angle X at the centre

Arc AB subtends angle Y at the circumference

Chord AB subtends angle X at the centre

Chord AB subtents angle Y at the circumference

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## Theorom 1

The angle subtended at centre of a circle is twice the angle subtended at the circumference by the same arc or chord.

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## Theorem 2

Angles in semi cricle are right angles

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## Theorem 3

Angles subtended by an arc or chord in the same segment are equal.

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## Therom 4

Angle between a tangent and a radius is 90 degrees

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## Theorem 5

Alternating segment theorem

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## Theorem 6

The oppersite angles of a cyclic quadrilateral are supplementary (add up to 180)

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## Arcs and Sectors

The angle between 2 radii of a circle divide the circle into a minor and a major sector.

The arc lenths of each sector are the minor and major arcs.

Arc length = centre / 360 x 2 pie R

Area of sector = centre / 360 x pie R squared

C = pieD

A = pie R squared

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## Volume of Prisms

Volume of a cuboid = length x width x height

Volume of a prism = area of cross section x length

Volume of a cylinder = pie R squared x H

1m cubed = 1000 litres

1cm cubed = 1000 milimetres

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## Surface Area

Surface area of a cylinder = 2pieRsqaured + 2pieRH

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## Simular Triangle

The 2 conditions for simularity between shapes are =

- corresponding sides are in proportion

- corresponding angles are equal

Triangles are the exception of the rule. Only the second condition is needed.

2 triangles are similar if their corresponding angles are equal.

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## Shapes

Congruent shapes = are identical in shape and size but may have different orientations (reflected, rotated etc)

Similar shapes = are identical in shape but different in size.

Triangles are similar if they are exactly the same shape but are a different size.

They will have identical angles and different length sides. These lengths will be the same proportion.

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Factorising - one the equation is factorised use the fact that one of the factors must be 0.

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Area = 1/2 BH

SinA = O/H

CosA = A/H

TanA = O/A

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## Sine Rule

Used to find =

- an unknown side when we are given two angles and a side

- an unknown angle when we are given two sides and an angle

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## Cosine Rule

Used to find =

- an unknown side when two sides and the included angle are given

- an unknown angle when 3 sides are given

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## Using Sine to find area of a triangle

Area = 1/2 abSinC

SinA = O/H

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## Bearings

- Draw a straight line joining the points.

- Draw a north line at the line bearing from.

- Measure the angle.

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