Slope is a measure of the steepness of a line (or segment) as well as an indicator as to whether the line is rising or falling.

The symbol for slope is "m".

Slope can be calculated by taking any two locations along a line and calculating the ratio of the RISE to the RUN (the ratio of the difference between vertical positions of the locations to the difference between horizontal positions of the locations).

Slope is a measure of the rate at which vertical position is changing per unit of horizontal change.

• m = 0: A horizontal line.
• m > 0: A line that is rising (left to right).
• m < 0: A line that is falling (left to right).

• m = 1: A line at 45°.
• m = -1: A line at 135° (or -45°}.
• |m| < 1 (i.e. -1 < m < 1): A line that is shallow.
• |m| > 1 (i.e. m < -1 or m > 1): A line that is steep.

Together, the information is as follows:

• m > 1: A line that is steep and rising.
• m = 1: A line that is rising at 45°.
• 0 < m < 1: A line that is shallow and rising.
• m = 0: A horizontal line.
• -1 < m < 0: A line that is shallow and falling.
• m = -1: A line that is falling at 135°.
• m < -1: A line that is steep and falling.

For more details, click here.

By setting "RUN" to a value of one, slope can be represented as a directed distance. The resulting directed distance determines whether the line rises or falls as well as how steep it is.

In the previous applet (above), drag one of the segment endpoint so that the "run" is set to one, then use the "up" and "down" arrow keys to move the other location vertically

In the applet below, drag the indicated point vertically along the m-axis.

Notice that in both cases, the value of the slope is equal to the value of the "rise" (when the value of the "run" is equal to one).

The slope of a vertical line is "undefined".

This occurs since the "run" is equal to zero, and division by zero is undefined.

Using either of the previous applets, drag a location so that the line (or segment) becomes steeper and steeper, resulting in the line becoming closer and closer to being vertical.

If the line approaches vertical from the counter-clockwise direction, the slope gets larger and larger, approaching positive infinity.

If the line approaches vertical from the clockwise direction, the slope gets smaller and smaller, approaching negative infinity.

If two lines are parallel, then their slope are equal.

If two lines are perpendicular, then their slopes are negative reciprocals.

The slope of any straight line is a constant. This constant slope is represented by a set of similar triangles all having the ratio of rise to run equal to the constant slope.

If θ is chosen so that it is always associated with a horizontal ray pointing from the line to the right, then the RISE is always represented by the opposite side and the RUN is always represented by the adjacent side. As a result, the relationship between the slope of a line and the angle the line makes with the ray is given by tan(θ) = m [or θ = tan^-1(m)].

EXAMPLES

Given the slope of the line, what is angle the line makes with horizontal?

EXAMPLES

Given the slope of the line, what is angle the line makes with horizontal?