Distance Between Two Points
In IB Mathematics, the length of the line joining the points (x1, y1) and (x2, y2) is:
Example
Find the distance between the points (5, 3) and (1, 4).
(So in this case, x2 = 1, x1 = 5, y2 = 4 and y1 = 3).
Distance = √ (1-5)2 + (4-3)2 = √17
The Midpoint of a Line Joining Two Points
The midpoint of the line joining the points (x1, y1) and (x2, y2) is:
(x1+x2)/2 + (y1+y2)/2
Example
Find the coordinates of the midpoint of the line joining (1, 2) and (3, 1).
Midpoint = (1+3)/2 + (2+1)/2= (2, 1.5)
The Gradient of a Line Joining Two Points
In IB Mathematics, the gradient of the line joining the points (x1, y1) and (x2, y2) is:
y2 - y1/ x2 - x1
Example
Find the gradient of the line joining the points (5, 3) and (1, 4).
Gradient = (4-3)/ (1-5) = -0.25
Parallel and Perpendicular Lines
If two lines are parallel, then they have the same gradient.
In IB Mathematics, if two lines are perpendicular, then the gradients of the two lines are reciprocals of each other.
Example
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
In IB Mathematics, the equation of a line which has gradient m and which passes through the point (x1, y1) is:
y - y1 = m(x- x1)
Example
Find the equation of the line with gradient 2 passing through (1, 4).
y - 4 = 2(x - 1)
y - 4 = 2x - 2
y = 2x + 2
Since m = y2 - y1/x2 - x1
In IB Mathematics, the equation of a line pass through(x1, y1) and (x2, y2) can be written as:
This is the end of this topic.