Maths
Module 5
- Created by: Laura Godfrey
- Created on: 19-04-10 11:41
Straight line graphs
General equation -- y = mx + c
m = gradient
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.
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.
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
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 )
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.
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
Theorom 1
The angle subtended at centre of a circle is twice the angle subtended at the circumference by the same arc or chord.
Theorem 2
Angles in semi cricle are right angles
Theorem 3
Angles subtended by an arc or chord in the same segment are equal.
Therom 4
Angle between a tangent and a radius is 90 degrees
Theorem 5
Alternating segment theorem
Theorem 6
The oppersite angles of a cyclic quadrilateral are supplementary (add up to 180)
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
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
Surface Area
Surface area of a cylinder = 2pieRsqaured + 2pieRH
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.
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.
Solving quadratic equations
Factorising - one the equation is factorised use the fact that one of the factors must be 0.
Area of triangles
Area = 1/2 BH
SinA = O/H
CosA = A/H
TanA = O/A
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
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
Using Sine to find area of a triangle
Area = 1/2 abSinC
SinA = O/H
Bearings
- Draw a straight line joining the points.
- Draw a north line at the line bearing from.
- Measure the angle.
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