Straight line graphs
General equation -- y = mx + c
m = gradient
c = y intercept (where the line cuts the y axis)
If 2 lines cross at a right angle the gradients are the same but with different signs.
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.
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
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
The angle subtended at centre of a circle is twice the angle subtended at the circumference by the same arc or chord.
Angles in semi cricle are right angles
Angles subtended by an arc or chord in the same segment are equal.
Angle between a tangent and a radius is 90 degrees
Alternating segment theorem
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 of a cylinder = 2pieRsqaured + 2pieRH
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.
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
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
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
- Draw a straight line joining the points.
- Draw a north line at the line bearing from.
- Measure the angle.