The Straight Line - Part 2

  y - b = m(x - a),  where (a, b) are co-ordinates of a point on the line.

Proof

Example

 

Gradient and Tangent

Gradient = tanθ    where θ is measured anticlockwise.

Proof

 

Example

What is the gradient of the line shown:

Or, if using Exact values

 

Example

 

 

Differentiation and gradient

Distance Formula

 This is used to find the length of the line between two points.

    

  

 

Example

An isosceles triangle has two sides of equal length.

Use the distance formula to discover the length of each side.

 

Midpoint Formula

 

 

Example

 

Find the co-ordinates of

• G the midpoint of BC
• F the midpoint of CD
• The midpoint of AC
• The midpoint of BD

What do you notice about your
answers for the last two quaestions  ?

The diagonals intersect at their midpoint.

Perpendicular lines

 

OA is perpendicular to OB

 

Example

Are the lines y=3x+7 and 3y + x = 54 perpendicular ?

 

 

Example

 

  

Start by finding the gradient of AB.

Then use this to findthe gradient of CD.

Now find the equation of line CD

Now find t.

 

collinearity

Points that sit on the same line are collinear.

Example

Example

Are the points A(-2,1) , B(-1,0) and C(7,-8) collinear?

It is not enough to just show equal gradients, since this just indicates that the lines are parallel. To be collinear, they must share an equal point.

General equation of a straight line

Example