Coordinate Geometry
Grids
To make a grid, we draw two lines, a horizontal line and a vertical line. These are called axes.
Midpoint of a Line Segment
Given diagram shows the point C(3,2) and D(5,4). If we join C to D, we get the line segment [CD]. M is the midpoint of [CD]. From the grid we can see that the coordinates of M are (4,3).
The Slope of a Line
To measure how steep a line or a line segment is we can use the idea of a mathematical slope. The slop is calculated using the rise over run formula.
Rising lines have positive slopes and falling lines have negative slopes.
Distance Between Two Points
Line Formula
Sometimes you could be given the slope and one point on a line and be expected to determine the equation of that line in the form y=mx+c where m = slope and c = y-intercept. This can be done using the following formula.
Equation of a line with Slope m and Containing (x1, y1):
l is the line containing the point (x1,y1) and (x,y) is any other point on the line l. m is the slope of line l. y-y1 = m(x-x1). This is the equation of the line. Therefore, to find the equation of a line you need the slope of the line, and a point on the line.
Axes Intercept
The y-intercept is where the line touches the y axis. The x-intercept is where the line touches the x axis. where a line cuts the x-axis, the value of the y-coordinate is 0. Where a line cuts the y-axis the value of the x-coordinate is 0.
Point of Intersection Between Two Lines
Where a line cuts the x-axis the value of the y coordinate is 0
Where a line cuts the y-axis the value of the x coordinate is 0
There are two methods for calculating the point of intersection between two lines.
Method 1: Graphing the two lines
The first method is to graph the two lines and read the point of intersection from the graph.
Method 2: Simultaneous Equations
Step1:
The first step is to find m or the slope of the line.
Step 2:
The second step is to solve for y by subbing in what we know to the line formula.
Step 3:
Solve for x by subbing in y and rearranging.
Reciprocal
Swap the top and bottom of the number in fraction form
Parallel and Perpendicular Lines
If two lines are parallel then they have equal slopes. Excluding the exceptional case of a vertical line and a horizontal line, if two lines are perpendicular then the product of their slopes is -1. In other words m1 x m2 = -1.
Graphing Equations and Functions
Lines parallel to the x-axis or y-axis
A line parallel to the y-axis has an equation of the form =m, where m is the same constant.
Lines of the form y=mx+c have slope m and y-intercept (0,c).
