Co-ordinate Geometry

Co-ordinate GeometryStraight line Gradient form: y = m + b General form: A + By + C = 0

Distance: d = (1-2)2 + (y1-y2)2

Midpoint: mp = (1 + 2 , y1 + y2 ) 2 2

Gradient: m = y2-y1 2-1

Perpendicular distance

P = A1 + By1 + C

A2 + B2

m = tan

Geometrical Properties Complementary angles add to 90 Supplementary angle add to 180 Vertically opposite angles are equal Angles at a point add to 360 Angle sum of a triangle is 180 The exterior angle of a triangle is equal to the sum of the opposite interior angles An isosceles triangle has equal base angles Equilateral triangles have all angles 60 Alternate angles on parallel lines are equal Corresponding angles on parallel lines are equal Co-interior angles between parallel lines are supplementary The angle sum of a polygon is (n-2)x180 The sum of the exterior angles of any polygon is equal to 360 The angle sum of a quadrilateral is 360

Tests for special quadrilaterals:Parallelograms: Two opposite sides equal and parallel or Opposite sides are equal or Opposite angles are equal or Diagonals bisect each otherRhombus: All sides equal or Diagonals bisect each other at right anglesRectangle: All angles are right angles or Parallelogram with equal diagonalsSquare: All sides equal and one angle right or All angles right and two adjacent sides equal.Tests for congruent triangles SSS SAS AAS RHSTests for similar triangles AA Corresponding sides proportional (SSS) Two sides are proportional and included angles are equal (SAS)

Applications of Differentiation First derivative dy/d- Stationary point when equals 0- Curve increasing>0- Curve decreasing0-Concave down when 1

Sn = if r < 1

S = where < 1Compound Interest

A = P

SuperannuationIf $P is invested at the beginning of each year in a superannuation fund earning interest at r% pa, the investment after n years will amount to T

A1 = P

A2 = P

And so on, so that investment = A1 + A2= P + P

forms a geometric series with

a = P n = number of years and r =

Time paymentsA person borrows $P at r% per term, where the interest is compounded per term on the amount owing. If they pay off the loan in equal term instalments over n terms, their equal term instalment is M, where

M =

Deriving the equation:An = P (rate)n M (1 + rate + rate2)

After fully paid An = 0Rearrange to find M, using (1 + rate + rate2) as a geometric series.

ProbabilityProbability of an event occurring =

The probability of two events A and B occurring is given by:P(AB) = P(A) x P(B)

Sum and Difference of Two CubesX 3+ Y 3 = (x + y)(X 2 - XY + Y 2 ) X 3 -Y 3 = (X - Y)(X 2 + XY + Y 2 )

Parabolas(-b)2 = 4a(y-c)where (b,c) is the vertexa is the focal length

1