The Gradient of a Line Joining Two Points
Apply the following formula when finding the gradient of the line joining the points (x1, y1) and (x2, y2):
y2 - y1/ x2 - x1
Find the gradient of the line joining the points (3, 2) and (4, 6).
Gradient = 6 - 2/4 - 3= 4/1 = 4
The Midpoint of a Line Joining Two Points
When finding the midpoint of the line joining the points (x1, y1) and (x2, y2) apply the following formula:
[½(x1 + x2), ½(y1 + y2)]
Find the coordinates of the midpoint of the line joining (3, 2) and (5, 3).
Midpoint = [½(5 + 3), ½(2 + 3)] = (4, 2.5)
Parallel and Perpendicular Lines
When two lines are parallel, they have the same gradient.
When two lines are perpendicular, the product of the gradients of the two lines are -1.
a) y = 2x + 1
b) y = -½ x + 2
c) ½y = x - 3
The gradients of the lines are 2, -½ and 2 respectively. Therefore (a) and (b) and perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.
The Equation of a Line Using One Point and the Gradient
The equation of a line which has gradient m and which passes through the point (x1, y1) is:
y - y1 = m(x - x1)
The equation of a line passing through (x1, y1) and (x2, y2) can be written as:
y - y1 = y2 - y1
x - x1 x2 - x1