Straight Lines

In geometry, a straight line is defined as a line segment that connects two points and extends infinitely in both directions. A straight line is the shortest distance between two points. The straight line is also considered to be the most basic type of line.

The properties of a straight line are:

  • A straight line is the shortest distance between two points.
  • A straight line extends infinitely in both directions.
  • A straight line is considered to be the most basic type of line.
  • Straight lines do not have any curves or bends.
  • Straight lines are one-dimensional.
  • A straight line does not have area or volume.

In geometry, we use straight lines to create shapes and figures. We can also use them to solve problems. For example, if we want to find the length of a side of a triangle, we can use a straight line to measure it. We can also use straight lines to bisect angles and segment segments.

Types of Straight Lines

Types of
Straight Lines		 Description Illustration
Horizontal Line Lines that extend from left to right or right to left horizontally are called horizontal lines. Horizontal lines are parallel to the y-axis and perpendicular to the x-axis Horizontal Line
Vertical Line Lines that extend from top to bottom or bottom to top vertically are called vertical lines. Vertical lines are perpendicular to x-axis and parallel to y-axis. Vertical Line
Oblique or
Slant Line		 Lines that are drawn slantingly are called oblique lines. Oblique lines are also called slanting lines. Oblique or Slant Line
Line Segment The infinite length extending in both directions between two points is called a line segment. Line Segment
Ray A line starting from one point and extending infinitely in the other direction is called a ray Ray
Parallel Lines Lines that extend on either side without meeting each other at a constant distance are called parallel lines. Parallel Lines
Intersecting Lines Lines that pass through each other are called intersecting lines. The point at which the lines pass is called the point of intersection. Intersecting Lines
Perpendicular Lines Two lines intersecting each other at an angle of 90° are called perpendicular lines. Perpendicular Lines
Read more about Types of Straight Lines at Careers360

 

Straight Line Formulae

Standard Form of a Linear Equation

In this formula, x and y are variables, while a, b, and c are constant numbers. Usually, a is positive and a, b, and c are integers. Also, a, b, c ∈ ℝ, both a ≠ 0 and b ≠ 0.

ax+by+c=0ax+by+c=0

Slope Between Two Points

The idea of slope is something you encounter often in everyday life. Think about rolling a cart down a ramp or climbing a set of stairs. Both the ramp and the stairs have a slope. You can describe the slope, or steepness, of the ramp and stairs by considering horizontal and vertical movement along them. In conversation, you use words like “gradual” or “steep” to describe slope. Along a gradual slope, most of the movement is horizontal. Along a steep slope, the vertical movement is greater. In math, slope is used to describe the steepness and direction of lines.

m=tan⁡(θ)=y2−y1x2−x1m\,\,=\,\,\tan \left( \theta \right) \,\,=\,\,\frac{y_2-y_1}{x_2-x_1}

Provided x1 ≠ x2

Lumen Learning

Slope of a Line

From the standard form of a line ax + by + c, and b ≠ 0

m=−abm=-\frac{a}{b}

 

Lumen Learning

Slope Intercept Form

You’re using a ride-share app that charges a $2 base fee plus $1.50 per mile. What equation can you use to predict your fare—and how can the graph of this equation help you compare trips? Before GPS was common, airline pilots used linear equations—often in slope-intercept form—to chart flight paths and adjust for wind drift. By plotting lines quickly and understanding their direction, they could navigate long distances using just a map, compass, and some math.

The most common formula for a straight line is the slope-intercept form: y = mx + b or y = mx + c. In this equation, y is the vertical coordinate, x is the horizontal coordinate, m is the slope or steepness of the line, and b or c is the y‑intercept where the line crosses the vertical axis.

y=mx+cy=mx\,\,+\,\,c

 

Figure 11.2.1. The line y = 2x + 1, with its y-intercept
(O, 1) marked and a slope triangle showing a rise of
2 over a run of 1

Point-Slope Form

Linear relationships are found in various real-world situations, such as determining the speed of a moving object, predicting financial growth, and analysing population trends. When we know a point on a line and the rate at which it changes, we can express the line using the point-slope form of an equation.

Unlike the slope-intercept form, which requires knowing the y — intercept, the point-slope form is useful when we have a point (Cl , YI) and the slope m. This form is particularly beneficial in cases where data is given in tabular form or obtained from experimental measurements.

Line through (x1, y1) with slope m

y−y1=m(x−x1)y-y_1 = m\left( x-x_1 \right)
Figure 3.4.1. A line through (x0, y0) with slope m.

Two-Point Form

Linear relationships occur in various real-world scenarios, such as determining the speed of a moving object, analyzing population trends, and modelling financial growth. The equation of a line provides a mathematical way to describe these relationships. When we know two points on a line, we can determine its equation using the two-point form, and it builds the slope directly into the equation so you can find the line in one step.

Unlike the slope-intercept form, which requires the y-intercept, the two-point form is particularly useful when only two distinct points on the line are given. This method is commonly used in physics, engineering, and computer graphics, where data is often obtained from experimental measurements or observations.

Two point form is one of the important forms used to represent a straight line algebraically. The equation of a line represents each and every point on the line, i.e., it is satisfied by each point on the line. The two-point form of a line is used for finding the equation of a line given two points (x1, y1) and (x2, y2) on it.

Given a line through and (x1, y1) and (x2, y2), provided x1 ≠ x2

y−y1=y2−y1x2−x1(x−x1)y-y_1=\frac{y_2-y_1}{x_2-x_1}\left( x-x_1 \right)

Intercept (or Two-Intercept) Form

You use the intercept form (or two-intercept form) of a linear equation when you know both the x-intercept and the y‑intercept of a line. It is very fast to use when you want to draw a graph on a coordinate plane, because you can immediately plot the points (a,0) and (0,b) and connect them with a straight line.

It is helpful in word problems or geometry applications where the focus is strictly on where a boundary or line meets the axes (such as finding intercepts or calculating areas of right triangles formed by the axes and the line).

xa+yb=1\frac{x}{a}+\frac{y}{b}=1

 

where a is the x-intercept and b is the y-intercept. The x-intercept is the x-coordinate of the point where the line crosses the x-axis, the point (a,0). The y-intercept is the y-coordinate of the point where the line crosses the y-axis, the point (0,b).

Condition for Perpendicular Lines

Two lines are perpendicular if and only if their slopes are opposite reciprocals.

m1=−1m2m_1=-\frac{1}{m_2}

or

m1⋅m2=−1m_1\cdot m_2\,\,=\,\,-1


See Why a 90 Degree Rotation Turns Every Nonzero Slope Into Its Negative Reciprocal for more information.

Condition for Parallel Lines

Two lines are parallel if and only if they are both vertical or they have the same slope.

m1=m2m_1=m_2

Angle Between Two Lines

The angle between is the measure of the inclination between the two lines. For two intersecting lines, there are two types of angles between the lines, the acute angle and the obtuse angle. Here we consider the acute angle between the lines to be the angle between two lines.

tan⁡(θ)=|m2−m11+m1m2|\tan \left( \theta \right) =|\frac{m_2-m_1}{1+m_1m_2}|

 

CUEMATH

References

“13.2.1: Finding the Slope of a Line.” LibreTexts, June 23, 2021. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/13%3A_Graphing/13.02%3A_Slope_and_Writing_the_Equation_of_a_Line/13.2.01%3A_Finding_the_Slope_of_a_Line.

“Angle Between Two Lines.” ALLEN, September 13, 2024. https://allen.in/jee/maths/angle-between-two-lines.

“Angle Between Two Lines – Formula, Examples, Tan, Cos.” CUEMATH. Accessed August 24, 2026. https://www.cuemath.com/geometry/angle-between-two-lines/.

Arnold, David. “3.4: The Point-Slope Form of a Line.” LibreTexts, May 15, 2019. https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/03%3A_Linear_Functions/3.04%3A_The_Point-Slope_Form_of_a_Line.

Bourne, Murray. “What Is a Straight Line in Geometry?” Interactive Mathematics. Accessed August 24, 2026. https://www.intmath.com/functions-and-graphs/what-is-a-straight-line-in-geometry.php.

“Equation of a Straight Line – Formulas and Examples.” Math Monks, February 26, 2025. https://mathmonks.com/equation-of-a-straight-line.

“Equation of Straight Line – Formula, Forms, Examples.” CUEMATH. Accessed August 24, 2026. https://www.cuemath.com/geometry/straight-line/.

“Equation of a Straight Line.” GeeksforGeeks, February 20, 2022. https://www.geeksforgeeks.org/maths/equation-of-a-straight-line/.

Khandelwal, Neha. “Point-Slope Form Equation of a Line.” CK-12 Foundation, April 24, 2026. https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-11/section/9.4/primary/lesson/point-slope-form-equation-of-a-line/.

Khandelwal, Neha. “Two-Point Form Equation of a Line.” CK-12 Foundation, May 25, 2026. https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-11/section/9.5/primary/lesson/two-point-form-equation-of-a-line/.

“Line (Geometry).” Wikipedia, July 30, 2026. https://en.wikipedia.org/wiki/Line_(geometry).

[ ℰ ] Miglani, Komal. “Straight Lines – Definition, Equations, Properties and Examples.” Careers360, October 24, 2024. https://www.careers360.com/maths/straight-lines-chapter-pge.

“Parallel and Perpendicular Lines.” Saylor University. Accessed August 24, 2026. https://learn.saylor.org/mod/book/tool/print/index.php?id=87743&chapterid=83162.

Roberts, Donna. “Perpendicular Lines.” MathBitsNotebook. Accessed August 24, 2026. https://mathbitsnotebook.com/Geometry/Equations/EQPerpendicularLines.html.

“Slope Intercept Form.” BYJU’S, March 19, 2020. https://byjus.com/maths/slope-intercept-form/.

“Straight Lines.” BYJU’S, March 3, 2019. https://byjus.com/jee/straight-lines/.

“Straight Line – Equations, Definition, Properties, Examples.” CUEMATH. Accessed August 24, 2026. https://www.cuemath.com/geometry/straight-line/.

“Straight Lines – Definition, Equations Formula, Solved Example Problems, Exercise | Analytical Geometry | Mathematics.” BrainKart. Accessed August 24, 2026. https://www.brainkart.com/article/Straight-Lines_33933/.

“Using the Slope Formula to Find the Slope between Two Points.” Developmental Math Emporium. Accessed August 24, 2026. https://courses.lumenlearning.com/wm-developmentalemporium/chapter/finding-slope-given-two-points-on-a-line/.

Videos

 

This geometry video tutorial provides a basic introduction into points, lines, segments, rays, and planes. It explains how to identify three collinear points and how to distinguish it from noncollinear points. It explains the difference between coplanar points and noncoplanar points. Three points are collinear if they lie on the same line. This video describes the four ways to determine a plane: 1. Three noncollinear points determine a plane. 2. Two parallel lines determine a plane. 3. Two intersecting lines determine a plane. 4. A point and a line can also determine it. This video also explains how to identify coplanar lines and noncoplanar lines and segments. It contains plenty of examples and practice problems.

 

 

This geometry video tutorial provides a basic introduction into lines, rays, line segments, points, and angles. It also explains the difference between the union and intersection symbols and contains a few practice problems associated with it. Lines extend infinitely in both directions. Rays have a common endpoint and extend infinitely in one direction. Line segments have a beginning and an end and possess a definite length. Angles are composed of two rays that meet at a common endpoint or a vertex.

[ ℰ ] This exceptional reference is highly recommended for your consideration.

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