straight line

• Different formulas for calculating the gradient

  m=Y2-Y1 m=tan0(theata) y=mx+c

  ------ , m=positive - 0=tan^-1 , m= gradient

  X2-X1 m=negative - 0=180-tan^-1
• • positive gradients slope up from left to right
  • negative gradients slope down from left to right
  • a line parallel to the x-axis has a gradient of 0
  • a line parallel to the -axis has an undefined gradient
  • parallel lines have the same gradient
• The formula for finding the midpoint of a line segment is given by (x1 + x2)/2 and (y1 + y2)/2.
• A perpendicular bisector is a straight line that passes through the middle point of a line segment, and it divides the line into two equal parts.
• rule for finding the gradient of the other line if it is perpendicular


  inverting the gradient and inverting the sign
• special case for perpendicular lines

  a line parallel to the x-axis is perpendicular to a line parallel to y-axis

  • M1 = 0 $->$ M2= underfined
• steps for finding the equation of a straight line

  • find gradient
  • find a + b
  • substitute into y-b=m(x-a) or y=mx+c
• Collinearity

  • when two lines are parallel and share a common point
  • they must touch

  
• a line parallel to the x-axis through P(a,b) has the equation


  y=b

  (draw it out)
• a line parallel to the y-axis through P(a,b) has the equation


  x=a

  (draw it out)
• the distance formula

  • work out the distance between two points
  • i.e. the length of a line

  
• midpoint formula

  • middle point between two co-ordinates

  
• Median


  • a line drawn within a triangle joining together one of the points with the mid-point of the opposite line

  steps for calculating median

  1. calculate the mid-point of the relevant line (BC below)
  2. calculate the gradient of the line between the mid-point and the opposite point (AD below)
  3. write down the equation using the gradient and either of the two points used in the second step (A or D below), and substitute into y-b=m(x-a)

  
• Altitudes


  • a line from a vertex of a triangle perpendicular to the opposite corner

  steps for the equation of the altitude

  1. calculate the gradient of the line which is perpendicular to the altitude (Ac below)
  2. calculate the gradient of the altitude using $m1$ x $m2$ =-1 (BD below)
  3. write down the equation using this gradient and the point that the altitude passes through (B below)
  4. substitute into y-b=m(x-a)

  
• • median goes through midpoint, but is not perpendicular
  • altitude doesn't go through midpoint, but is perpendicular
  • perpendicular bisector goes through midpoint and is perpendicular
• Perpendicular Bisector

  line which cuts through the middle of another line at a right angle

  steps for equation of a perpendicular bisector

  1. calculate the mid-point of the line that the perpendicular bisector bisects (XY below)
  2. calculate the gradient of the same line used in step one (XY below), then find the gradient of the perpendicular bisector (blue line below)
  3. write down equation using this gradient and the midpoint, which the perpendicular bisector passes through, substitue into y-b=m(x-a)

  
• Intersection lines

  • When two or more lines intersect (cross over each other)

  
  

  • find equations of lines (will be two out of median, altitude and perpendicular bisector)
  • calculate point of intersection by using simultaneous equations
  • for using simultaneous equations , may have to equate (set equal to each other if they have the same x/y co-efficient)

  or by substitution (nat5)