Line

A line is a straight, one-dimensional figure. The term one-dimensional means that the line has no thickness, only length. A line has no endpoints, meaning it goes on infinitely it goes on forever. A straight one-dimensional object which has one endpoint is called a ray. A straight, one-dimensional object which has two endpoints is called a line segment. See figure 1. An endpoint is a point on the end of a ray or line segment.

In Elements Euclid defined a straight line as, "A straight line is a line which lies evenly with the points on itself." This statement seems to confuse more than explain. This demonstrates the difficulty in defining base concepts. In modern Euclidean geometry, 'straight' means what we usually mean by the word straight. It goes on without curving. However, in spherical geometry, straight means that, though the line follows the curve of the surface of the sphere, as in figure 2, it does not turn to the left or the right.

The slope of a line is the ratio of the rise of the line divided by the run. The rise refers to the vertical distance between any two distinct points and run refers to the horizontal distance between the same two points. Slope is also called the rate of change. In cases of direct variation, the slope is also called the constant of variation.

How to Calculate the Slope of a Line

To calculate the slope of a line, first identify the coordinates of any two distinct points. The coordinates of the lower left point in figure 3 are (0, -1). Call this (x₁, y₁). The coordinates of the upper right point in figure 3 are (1, 1). Call this (x₂, y₂). It doesn't matter which point is called (x₁, y₁) and which is called (x₂, y₂). The answer will come out the same.

The formula for the slope is a = (y₂ - y₁)/(x₂ - x₁). Substituting the points from figure 3 into the formula, we get a = (1 - (-1)) / ( 2 - 0 ). Simplifying the numerator and denominator, we get a = 2 / 2 = 1. So the slope of the line in figure 3 is 1.

Understanding Check

Write the answers to the following problems on a piece of paper. Then click on the blue and yellow words to see the correct answer.

1. What is the slope of the line in figure 4? Calculate it on paper and then check your work by clicking on the symbols in yellow on a blue background.

   m = ( y₂5 - y₁1 ) / ( x₂2 - x₁0 ) = ( ?4 / ?2 ) = ?2.

The table in figure 5 shows a graph of the total cost of gasoline as a function of the number of gallons pumped. Read each question, and write your answer on a piece of paper.

1. If one gallon is pumped, what is the total cost of the gasoline? Click for Answer$3.00
2. If two gallons are pumped, what is the total cost of the gasoline? Click for Answer$6.00
3. For every gallon that is pumped, the total cost of the gasoline goes up how much? Click for Answer$3
4. If x gallons are pumped, what is the total cost of the gasoline? Click for Answer$3 · x
5. What is the rate of change for this relationship? Click for Answer3
6. What is the equation for this relationship? Click for Answery = 3x

In Euclidean geometry, two lines are parallel if they do not intersect⁵. In metric geometry, parallel lines have the same slope. Since the two lines have the same rate of change, for the same change in x the change in y will be identical. So the lines will always be the same distance apart and will never intersect.

Two lines intersect if they cross each other. Another way to look at it is; two lines intersect if they have exactly one point in common. In figure 7, the two lines intersect. Since figure 7 is two dimensional and the lines do not have the same slope, they have to intersect.

Properties of Intersecting Lines in Euclidean Geometry

• Opposite angles are congruent.
• The sum of any two adjacent angles is 180° = π radians.
• Intersecting lines intersect exactly once.

How to Find the Coordinates of the Intersection of Lines

To find the point at which two lines intersect, use substitution:

Two lines are perpendicular if they intersect at right angles (see Euclid. Elements Book 1 Definition 10. Translated by D. Joyce.). A right angle is 90° = π / 2 radians. In diagrams, right angles are denoted with a small square. See figure 9.

In metric geometry, you can tell if lines are perpendicular from the slopes. If m₁ is the slope of one line, and m₂ is the slope of a line perpendicular to the first, then m₁ = -1 / m₂.

Understanding Check

One paper, fill in the equation used to tell if lines are perpendicular using the equations y = 2x + 1 and y = -(1/2)x - 2. Are the two lines perpendicular? Write down your answer then click on the words below to see if your answer is correct.

m₁2 = -1/(m₂-1/2) = m₁2.
The two lines (are/are not)are perpendicular.

Skew lines are lines that do not intersect and are not parallel⁶.

In a two dimensional Euclidean plane, lines either intersect or are parallel, so skew lines do not exist in two dimensional space. Skew lines exist only in spaces with three or more dimensions.

Manipulative shows a pair of skew lines in a simulated three dimensional space. Click on the blue point and drag it to animate the space.

A line is vertical if it goes straight up and down. A vertical line is used for the y axis when graphing. The equation of a vertical line is in the form x = a where a is the x-intercept.

Horizontal lines go from side to side. One way to remember this is to remember that the horizon is horizontal. The equation of a horizontal line is used for the y = b where b is the y-intercept.

Slope-intercept form of a linear equation is y = mx + b. m is the slope of the line. b is the y-intercept of the line. A vertical line can not be represented in slope intercept form. Click on the points of the sliders in manipulative 2 and drag them to change the figure.

The point-slope form of a linear equation is y - y₁ = a (x - x₁), where (x₁, y₁) is any point on the line and a is the slope of a line. A vertical line can not be represented in point slope form.

For example, if the point (1, 2) is on a line with the slope 3, then the equation of the line can be written y - 2 = 3(x - 1).

A ray is a part of a line with one end point. In the other direction, the ray goes on forever, just like a line. Draw a ray with a dot at one end representing the end point and a line radiating from the other end. To show that one end of a ray goes on forever, draw an arrow (see figure 15). To write down a ray with endpoint A and a point B on the ray, Write with the arrow over the top. Another word that means the same thing as ray is half-line.

Coterminal rays are rays that share a common endpoint (see figure 16). Opposite rays are rays with an endpoint in common that go in opposite directions. Two rays are parallel if they are contained by the same line or lines that are parallel.