Line
Pronunciation: /laɪn/ Explain
 Figure 1: A line, ray and line segment 

A line is a straight, onedimensional
figure. The term onedimensional 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 onedimensional object which has one endpoint
is called a ray. A straight, onedimensional
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.

 Figure 2: A line in spherical geometry. 

Related Words
Article Contents
Properties of Lines in Euclidean Geometry
A line is uniquely determined by two points.
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. See
Euclid's Elements, Book 1, Postulate 1

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.
Equations of Lines
An equation that represents a line is called a
linear equation.
There are several customary forms of linear equations:
Vertical line passing through the xaxis
at (b,0).
ax + by = c  Standard form: This is used when
representing linear systems.
This form is also called the general form of a linear equation. 
y = ax  Direct variation:
This form is used when y changes in proportion with
x.

y = ax + b  Slope intercept form:
a is the slope of the line and
b is the yintercept. 
y  y_{1} = a(x  x_{1}) 
Point slope form: (x_{1},
y_{1}) is any point on
the line, a is the slope of the line. 
y = a  Horizontal line passing through the yaxis
at (0,a). 
x = b 
Slope of a Line
 Figure 3: Slope of y = x  1 
 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 (x_{1}, y_{1}).
The coordinates of the upper right point in figure 3 are (1, 1).
Call this (x_{2}, y_{2}). It doesn't matter
which point is called (x_{1}, y_{1}) and which
is called (x_{2}, y_{2}). The answer will come
out the same.
The formula for the slope is
a = (y_{2}  y_{1})/(x_{2}  x_{1}).
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.

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 =
( y_{2}5 
y_{1}1 ) /
( x_{2}2 
x_{1}0 ) =
( ?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).
Understanding Check
 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.
 If one gallon is pumped, what is the total cost of the gasoline? Click for Answer$3.00
 If two gallons are pumped, what is the total cost of the gasoline? Click for Answer$6.00
 For every gallon that is pumped, the total cost of the gasoline goes up how much? Click for Answer$3
 If x gallons are pumped, what is the total cost of the gasoline? Click for Answer$3 · x
 What is the rate of change for this relationship? Click for Answer3
 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 nonlinear equations.
Teachers, see also Rate of Change Lesson by Cynthia Lanius.
Parallel Lines
 Figure 6: Parallel Lines 

In Euclidean geometry, two lines are parallel
if they do not intersect^{5}.
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
 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:
Equation  Description 
y=x+1, y=2x1  Equations of the two lines 
2x1=x+1  Substitute 2x1
from the second equation for y in the first. 
3x1=1  Add x to both sides. 
3x=2  Add 1 to both sides. 
x=2/3  Divide both sides by 3. 
y=(2/3) + 1  Substitute 2/3 in for x on the first equation. 
y=1/3  Simplify 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
 Figure 8: Perpendicular Lines 
 Figure 9: Small square denoting a right angle. 

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
m_{1} is the slope of one line, and
m_{2} is the slope of a line perpendicular
to the first, then m_{1} = 1/m_{2}.
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/(m21/2)
= m12.
The two lines (are/are not)are perpendicular.

Skew Lines
Click on the blue point and drag it to change the figure.
Dragging the blue point rotates the figure. In some positions it seems like the two line intersect, and in others it does not. Why is this?
 Manipulative 1  Skew Lines Created with GeoGebra. 

Skew lines are lines that do not
intersect
and are not parallel^{6}.
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
 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
xintercept.

Horizontal Lines
 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
yintercept.

Slope Intercept Form of Linear Equations
Click on the blue points and drag them to change the figure.
Can you write the equation of a horizontal line in slopeintercept form? Can you write the equation of a vertical line in slopeintercept form?
 Manipulative 2  Line: SlopeIntercept Form Created with GeoGebra. 

Slopeintercept form of a linear equation is
y = mx + b. m is
the slope
of the line. b is the
yintercept
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
Click on the blue dot on the slider and on the line to change the figure.
Can you put a horizontal line in pointslope form? Can you put a vertical line in pointslope form?
 Manipulative 3  Line: PointSlope Form Created with GeoGebra. 

The pointslope form of a linear equation
is y  y_{1} = a ( x  x_{1} ), where
( x_{1}, y_{1} ) 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
 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
with the arrow over the top. Another word that means the same thing as ray is
halfline.
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.

 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
button in the manipulative window. To show the changes
for each step, click on the 'show' button for each step.
Click on the blue points and drag them to change the figure.
 Manipulative 4  Copy a Line Segment Created with GeoGebra. 


More Information
 Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.
Cite this article as:
McAdams, David E. Line. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/line.html.
Image Credits
Revision History
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
8/31/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (
McAdams, David E.)
10/11/2010: Added coterminal, parallel, and halfline to Rays section. (
McAdams, David E.)
12/5/2008: Added formal definition of endpoint. (
McAdams, David E.)
11/19/2008: Changed manipulatives to GeoGebra. Added javascript support for construct a line segment manipulative (
McAdams, David E.)
6/7/2008: Corrected spelling errors. (
McAdams, David E.)
3/11/2008: Corrected bad link for ray. (
McAdams, David E.)
8/28/2007: Add reference to Euclid's. (
McAdams, David E.)
8/22/2007: Add Article Contents. (
McAdams, David E.)
8/21/2007: Add construction of a line given a line and an endpoint. (
McAdams, David E.)
8/14/2007: Initial version. (
McAdams, David E.)