Coordinate Geometry-The Line - created from Mind Map

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Coordinating the plane

The horizontal line is called the x-axis.

The vertical line is called the y-axis.

The point (0,0) is called the origin and is noted O.

This coordinate plane is called the Cartesian plane.

Distance between two points

The distance between two points, A(x1,y1) and B(x2,y2) is given by the formula |AB|=((x2-x1)^2+(y2-y1)^2)^1/2.

Equidistant means the same distance.

The midpoint of a line segment

The midpoint of the line segment joining A(x1,y1), and B(x2,y2) is ((x1+x2)/2 , (y1+y2)/2).

The slope of a line

The slope of a line is defined as

1. The vertical change divided by the horizontal change.

2. Rise divided by Run.

The slope, m, of the line passing through (x1,y1) and (x2,y2) is m=(y2-y1)/(x2-x1).

Positive and negative slopes

Parallel lines

Parallel lines have equal slopes.

Perpendicular lines

If two lines are perpendicular, the product of their slopes is -1. i.e. m1*m2=-1

The equation of a line

The equation of the line through (x1,y1) with slope m is found by using y-y1=m(x-x1).

Equation of a line when given two points on the line

You may use either of the two points as(x1,y1).

The equation of y=mx+c

If the equation of a line is in the form y=mx+c, then

1. The slope is m.

2. The line intersects the y-axis at (0,c).

Parallel and perpendicular lines

If we are given the equation of a line l, such as 2x+3y-4=0, we can find the slope of the line by expressing the equation in the form y=mx+c.

If we are also given a point (x1,y1), we can then find the equation of a line through (x1,y1) and which is parallel to or perpendicular to l.

Graphing lines

To draw a line we need to know at least two points on the line.

The easiest points to find are those at which the line crosses the x-axis and the y-axis.

e.g. The line 2x+3y=6

when x=0

2(0)+3y=6

=> 3y=6

=> y=6/3

=> y=2

N.B. (0,2)

when y=o

2x+3(0)=6

=> 2x=6

=> x=6/2

=> x=3

N.B. (0,3)

Lines parallel to the axes

All lines with equations in the form x=a will be parallel to the y-axis.

All lines with equations in the form y=a will be parallel to the x-axis.

Lines containing the origin

A line such as x+2y=0, with no independent term, always contains the origin (0,0).

To verify that a point is on a given line

If a point is on a line, then the coordinates of the point will satisfy the equation of the line.

Intersection of two lines

Simultaneous equations can be used to find the point of intersection of two lines.

Area of triangle

For a triangle with vertices (0,0), (x1,y1) and (x2,y2), the application of this formula 1/2|x1*y2-x2*y1| results in finding the area of the triangle.