Explanation for the coordinate geometry of a straight line
Every straight line can be represented algebraically in the form y = mx + c , where
m represents the gradient of a line (its slope, steepness)
c represents the y -intercept (a point where the line crosses the y axis)
Furthermore, there are several ways in which you can describe a straight line algebraically
Equation of a line
The gradient measures the steepness of the line.
It is defined as , or
When the gradient is 1, the line makes a 45 0 angle with either axes. If the gradient is 0, the line is parallel to the x axis.
Equation of a straight line given the gradient and a point
If the point is given by its coordinates , and the gradient of a line is given as m , you can deduce the equation of that line.
You are using the formula for gradient, , to derive a formula for the line itself. The coordinates of the point can be substituted, while the y 2 and x 2 need to remain (without the superscript numbers).
Then simply substitute the given values into
The equation of a line given two points
When you have this kind of problem, you take that, as both points belong to the same line, the gradients at both points will be the same.
It makes sense therefore to say that
All you need to do in this case will be to substitute coordinates you have for the given points and .
Parallel and perpendicular lines
When two lines are parallel, their gradient is the same:
When two lines are perpendicular, their product equals -1: .
The line length
The length of the line segment joining two points will relate to their coordinates. Have a good look at the diagram
The length joining the point A and C can be found by using Pythagoras' Theorem:
Mid-point of a line
Mid-point of the line can be found by using the same principle
So the point between A and C will have the coordinates
If you know the midpoint, you can easily find the perpendicular bisector of a given line. This new line will go through the midpoint of the given line, and it will be perpendicular to it.
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