straight line

• Gradient Formulas


  m = y₂-y₁/x₂-x₁

  m = tanθ
• Explanation of m=tanθ


  θ is the angle which the line makes with the positive direction (right side) of the x-axis
• Process for finding angle between a line and the positive direction of the x-axis


  step 1: find the gradient of the line using co-ordinates

  step 2: rearrange the equation to find the angle

  step 3: sub in the gradient (if the gradient is negatibe sub in the positive value and do θ=180-tan⁻¹(m))
• State the equation of a line when the gradient is undefined


  x = value of x given (this will be the same in all co-ordinates)
• State the equation of a line when the gradient equals 0


  y = value of y given (this will be the same in all co-ordinates)
• Process of finding the gradient of a perpendicular line


  Step 1: find the gradient of the first line

  Step 2: flip the fraction and change the sign in front of the first gradient

  ex. mₓᵧ = 4 becomes m⊥=-1/4

  step 3: state the answer in the form of m⊥=... and then state 'since m₁ x m₂= -1 for perpendicular lines
• Define median


  line which joins a vertex (corner) to the mid-point of the opposite side

  
• State the process of finding the median


  Step 1: identify the midpoint (point 2&3 are the non-opposite vertices)

  Step 2: identify the gradient (point 1 is midpoint and point 2 is vertex)

  Step 3: identify the equation using the gradient and one of the points

  Step 4: state the equation in general form
• State the midpoint formula


  (x₂+x₁)/2, (y₂+y₁)/2
• Define altitude


  perpendicular height of a triangle from a vertex to meet the opposite side at a right angle

  
• State the process of finding the altitude


  Step 1: identify the gradient of the line perpendicular to the altitude

  Step 2: find the perpendicular gradient in order to find the gradient of the altitude

  Step 3: use the vertex to find the equation

  Step 4: state in general form
• Define perpendicular bisector


  line which cuts through the midpoint of a line at a right angle
• State the process of finding the perpendicular bisector


  Step 1: identify the midpoint of the line which is being cut

  Step 2: calculate the gradient of the line which is being cut

  Step 3: identify the gradient of the perpendicular bisector by finding the perpendicular gradient

  Step 4: find the equation of the line by using the gradient of the perpendicular bisector and the midpoint

  Step 5: state in general form
• Define collinear


  points which lie on the same straight line
• State the three different possibilities for two lines to be similar


  They can be connected at an angle to each other

  They can be parallel to each other

  They can be collinear
• Process to test different points for collinearity


  Step 1: find the gradient of each of the points with the other ones

  Step 2: identify if the gradients are the same or not

  Step 3: if they are the same state 'since mₐₑ=mₑᵥ, AE and EV are parallel. Since both lines share the same point of E, the lines are collinear.

  Step 4: if they aren't the same state 'since mₐₑ≠mₑᵥ, AE and EV aren't parallel. These lines aren't collinear. Since they both share the same point of E, they are at an angle to each other.