The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure. The horizontal real number line is usually called the x-axis, and the vertical real number line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants. Each point in the plane corresponds to an ordered pair (x,y) of real numbers and called the coordinates of the point.

The distance formula is derived by using the Pythagorean Theorem. From the figure:

The distance between the points and in the coordinate plane is

The midpoint of the line segment joining the points and in the coordinate plane is

Example 1/ Find (a) the distance between and (b) the midpoint the line segment joining, the points and .Solution /

Homework 1/ Find (a) the distance between and (b) the midpoint the line segment joining, the points and .

An equation of the line passing through the point (and having sloped

Therefore we need point ( and the slope.

Notes: A horizontal line has slope zero because y = 0 . The slope of a vertical line is undefined because = 0.Example / Find an equation of the line passing through the point and having slope .Solution /

Example 3/ Find an equation of the line passing through the points and .Solution /

With and

The angle of inclination of a line that crosses the -axisis the smallest angle we get when we measure counter clock-wise from the -axis around the point of intersection. tanWhere is the angle of inclination.Notes:1) The angle of inclination of a horizontal line is taken to be 0.2) Parallel lines have equal angle of inclination

Example / (A) Find the slope of a line whose angle of inclination is 60 () (B) Find the angle of inclination of a line with slopeSolution /(A) (B)

A non vertical line crosses the y-axis at some point (. The number is called the y-intercept of the line

Which is called the slope-intercept form of an equation of a lineExample 4/ Find an equation of the line with slope and y-intercept 4.Solution /

With and

An equation of the form

Where , and C are constants and A and B are not both zero, is called a first-degree equation in and.NOTE: we can find the slope by comparing the given equation with equation of intercept line .Or from

Example / Find the slope of the line with equation.Solution /

The slope of the line and y-intercept OR

Two lines and with slopes and, respectively, are parallel if and only if

and only if they have the same angle of inclination

Two lines and with slopes and, respectively, are pendicular if and only if

Example / Find an equation of the line that passes through the point and is perpendicular to the line with equation.Solution /

The slope of the line

1