# Line

Pronunciation: /laɪn/ Explain Figure 1: A line, rayand line segment

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. Figure 2: A line inspherical geometry.

### 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 lines in the same plane either intersect once or are parallel.

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

• #### Equations of Lines

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

 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 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. y = a Horizontal line passing through the y-axis at (0, a). x = b Vertical line passing through the x-axis at (b, 0).

### 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.

#### 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.

#### 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.

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

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

### Parallel Lines 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 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.

#### How to Find 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 + 1 Substitute 2x - 1 from the second equation for y in the first.
3x - 1 = 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 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

 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 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 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 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

 Click on the blue points and drag them to change the figure. Can you write the equation of a horizontal line in slope-intercept form? Can you write the equation of a vertical line in slope-intercept form? Manipulative 2 - Line: 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

 Click on the blue dot on the slider and on the line to change the figure. Can you put a horizontal line in point-slope form? Can you put a vertical line in point-slope form? Manipulative 3 - Line: 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 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 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. Figure 16: Coterminal rays

### How to Construct 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.

StepDescriptionJustification
0 Start with a line segment and a point not on the line. A straight line with the length equal to BC will be constructed with the end point at A. These are the criteria.
1 Construct line segment AB. Euclid Elements Book 1 Postulate 1: A line segment can be draw between any two points.
2 Construct the equilateral triangle DAB. Euclid Elements Book 1 Proposition 1: An equilateral triangle can be constructed on any line segment.
3 Extend the line segment DA as ray DE. Euclid Elements Book 1 Postulate 2: A line segment can be extended infinitely.
4 Extend the line segment DB as ray DF. Euclid Elements Book 1 Postulate 2: A line segment can be extended infinitely.
5 Draw a circle with center B and radius BC Label the intersection of the circle and the ray DF as G. Euclid Elements Book 1 Postulate 3: A circle can be draw with any center and any radius.
6 Construct a circle with center D and radius DG. Label the intersection of line DE and the circle DG as L. Euclid Elements Book 1 Postulate 3: A circle can be draw with any center and any radius.
7 Since the point B is the center of the circle CG, BCBG. Euclid Elements Book 1 Definition 15-18: A circle is all points the same distance from the center of the circle.
8 Since the point D is the center of the circle GDL, DLDG. Euclid Elements Book 1 Definition 15-18: A circle is all points the same distance from the center of the circle.
9 Since DGDL and DBDA, then the remainder BG must be equivalent to AL. Euclid Element Book 1 Common Notion 3: If equals are subtracted from equals, the remainder is equal.
10 Since BCBG and ALBG, then BC must be equivalent to AL. So a line equivalent to BC has been placed with an endpoint at A.
Q.E.D.
Euclid Element Book 1 Common Notion 1: If A = B and B = C then A = C.

### Table of Figures

 Figure Description Figure 1 A line, ray and line segment Figure 2 A line in spherical geometry Figure 3 Slope of y = x - 1 Figure 4 Slope of y = 2x + 1 Figure 5 Rate of Change Figure 6 Parallel lines Figure 7 Intersecting lines Figure 8 Perpendicular lines Figure 9 Small square denoting a right angle Figure 10 Skew lines Figure 11 Vertical lines Figure 12 Horizontal lines Figure 13 Slope-intercept form Figure 14 Point-slope form Figure 15 Rays Figure 16 Coterminal Rays
1. McAdams, David E.. All Math Words Dictionary, line. 2nd Classroom edition 20150108-4799968. pg 108. Life is a Story Problem LLC. January 8, 2015. Buy the book

• Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.

McAdams, David E. Line. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/line.html.

### Revision History

4/24/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)