Line

Pronunciation: /laɪn/ ?
This image shows a line with no end points, a ray with one end point and a line segment with two endpoints.
Figure 1: A line, ray
and line segment

A line is a straight, one-dimensional figure. The term one-dimensional means that the line has no thickness, only length[1][3]. 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."[2] 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.

This image shows a sphere with lines on its surface.
Figure 2: A line in
spherical geometry.

Related Words

  • Linear: Having to do with a line.
  • Linear Equation: A linear equation represents a line.
  • Linear System: A linear system is a set of related linear equations.

Article Contents

Properties of Lines in Euclidean Geometry

  • A line is uniquely determined by two points.4

    The word "uniquely" means that two points can determine only one line. Any line that passes through the two points must be the same line.

  • Two lines in a plane either intersect once or are parallel5.

    Two straight lines can not intersect twice. If two figures intersect twice, at least one must curve.

  • Two lines in three or more dimensions intersect, are parallel, or are skew.6

  • Equations of Lines

    An equation that represents a line is called a linear equation. There are several customary forms of linear equations:

    ax + by = cStandard form: This is used when representing linear systems. This form is also called the general form of a linear equation.
    y = axDirect variation: This form is used when y changes in proportion with x.
    y = ax + bSlope intercept form: a is the slope of the line and b is the y-intercept.
    y - y1 = a(x - x1) Point slope form: (x1, y1) is any point on the line, a is the slope of the line.

Slope of a Line

Graph of y = x - 1 showing the slope is 1.
Figure 3: Slope of y = x - 1

Graph of y = 2x + 1 showing the slope is 2.
Figure 4: Slope of y = 2x + 1

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.

Calculating 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.

check mark 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.

Rate of Change for Lines

When relating linear equations to the real world, the term rate of change is often used. This means that given a change in the independent variable (x), one can apply the rate of change as a ratio to find the change in the dependent variable (y).

Check markUnderstanding Check

Total cost of gasoline as a function of number of gallons pumped: y = 3x.
Figure 5: Rate of Change

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. The equation for this relationship? Click for Answery = 3x

Note: The term "rate of change" has a similar, but not identical, meaning when applied to non-linear equations.

Teachers, see also Rate of Change Lesson by Cynthia Lanius.

Parallel Lines

Graph of y=-x + 1 and y=-x - 2 showing that the lines are parallel.
Figure 6: Parallel Lines

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.

Intersecting Lines

Graph of y=-x + 1 and y=2x - 1 showing that the lines intersect.
Figure 7: Intersecting Lines

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.

Finding the Coordinates of the Intersection of Lines

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

EquationDescription
y=-x+1, y=2x-1Equations of the two lines
2x-1=-x+1Substitute 2x-1 from the second equation for y in the first.
3x-1=1Add x to both sides.
3x=2Add 1 to both sides.
x=2/3Divide both sides by 3.
y=-(2/3) + 1Substitute 2/3 in for x on the first equation.
y=1/3Simplify the right hand side of the equation.
(x,y) = (2/3,1/3)The lines intersect at the point (2/3,1/3)

Perpendicular Lines

Graph of y=2x + 1 and y=-(1/2)x - 2 showing that the lines are perpendicular.
Figure 8: Perpendicular Lines
Small square denoting a right angle.
Figure 9: Small square denoting a right angle.

Two lines are perpendicular if they intersect at right angles7. 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.

check mark 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

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Manipulative 1: Skew lines. Created with GeoGebra.

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.

Vertical Lines

Vertical lines go straight up and down
Figure 11: Vertical Lines

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

Horizontal lines go from side to side.
Figure 12: Horizontal Lines

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 Linear Equations

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Manipulative 2: Slope intercept form. Created with GeoGebra.

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.

Point Slope Form of Linear Equations

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Manipulative 3: Point slope form. Created with GeoGebra.

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 ).

Rays

Examples of rays.
Figure 15: Rays

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 AB with a one sided arrow pointing to the right. 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.

Two rays with a common endpoint that go in different directions.
Figure 16: Coterminal rays

Constructing a Line Segment Given a Segment and an Endpoint

Manipulative 4 will help visualize Euclid's proof of constructing a line segment the same size as an existing line segment at an existing point. To change the manipulative at each step, click on the blue points and drag them. To reset the manipulative to its original condition click on the reset GeoGebra reset button button in the manipulative window. To show the changes for each step, click on the 'show' button for each step.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 4: Copying a line segment. Created with GeoGebra.

StepShowDescriptionJustification
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Table 1

Table of Figures

FigureDescription
Figure 1A line, ray and line segment
Figure 2A line in spherical geometry
Figure 3Slope of y = x - 1
Figure 4Slope of y = 2x + 1
Figure 5Rate of Change
Figure 6Parallel lines
Figure 7Intersecting lines
Figure 8Perpendicular lines
Figure 9Small square denoting a right angle
Figure 10Skew lines
Figure 11Vertical lines
Figure 12Horizontal lines
Figure 13Slope-intercept form
Figure 14Point-slope form
Figure 15Rays
Figure 16Coterminal Rays

Cite this article as:


Line. 2010-10-11. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/line.html.

Translations

Image Credits

Revision History


2010-10-11: Added coterminal, parallel, and half-line to Rays section (McAdams, David.)
2008-12-05: Added formal definition of endpoint (McAdams, David.)
2008-11-19: Changed manipulatives to GeoGebra. Added javascript support for construct a line segment manipulative (McAdams, David.)
2008-06-07: Corrected spelling errors (McAdams, David.)
2008-03-11: Corrected bad link for ray (McAdams, David.)
2007-08-28: Add reference to Euclid's Elements (McAdams, David.)
2007-08-22: Add Article Contents (McAdams, David.)
2007-08-21: Add construction of a line given a line and an endpoint (McAdams, David.)
2007-08-14: Initial version (McAdams, David.)

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