How to Write Equation of Parallel and Perpendicular Lines?

Parallel and perpendicular lines are an important part of geometry and have distinctive features that help to easily identify them. In this step-by-step guide, you will learn about defining parallel and perpendicular lines and writing their equations.

How to Write Equation of Parallel and Perpendicular Lines?

Perpendicular and parallel lines have one common characteristic between them. They both consist of straight lines.

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Step by step guide to parallel and perpendicular lines

If two straight lines are on the same plane and if they never intersect, they are called parallel lines. They are always the same distance apart and are equidistant lines. The symbol \(||\) is used to represent parallel lines. For example, \(AB || DF\) means line \(AB\) is parallel to line \(DF\). On the other hand, when two lines intersect at a \(90\)-degree angle, they are known as perpendicular lines. Perpendicular lines are denoted by the symbol \(⊥\). For example, \(PQ ⊥ RS\) means line \(PQ\) is perpendicular to line \(RS\). 

Properties of parallel lines

  • Parallel lines are always equidistant from each other.
  • They never meet at any common point.
  • They lie in the same plane.

Properties of perpendicular lines

  • Perpendicular lines always intersect at \(90°\).
  • All perpendicular lines can be called intersecting lines, but not all intersecting lines can be called perpendicular because they must intersect at right angles.

Difference between parallel lines and perpendicular lines

The difference between the parallel lines and perpendicular lines are given below:

Equations of parallel and perpendicular lines

The equation of a straight line is represented as \(y=ax+b\) which defines the slope and the \(y\)-intercept. Here \(a\) represents the slope of the line. Since two parallel lines never intersect and have the same slope, their slope is always equal. For example, if the equations of two lines are given as, \(y=-2x + 6\) and \(y=-2x – 4\), we can see that the slope of both the lines is the same \((-2)\). Therefore, they are parallel lines. Mathematically, this can be expressed as \(m_1= m_2\), where \(m_1\) and \(m_2\) are the slopes of two lines that are parallel.

Perpendicular lines do not have the same slope. The slope of one line is the negative reciprocal of the other line. This can be expressed mathematically as \(m_1 × m_2 = -1\), where \(m_1\) and \(m_2\) are the slopes of two lines that are perpendicular. For example, if the equations of two lines are given as \(y = \frac{1}{5}x + 2\) and \(y=- 5x + 3\), we can see that the slope of one line is the negative reciprocal of the other. So, they are perpendicular lines. In this case, the negative reciprocal of \(-5\) is \(\frac{1}{5}\) and vice versa. Negative reciprocal means, that if \(m_1\) and \(m_2\) are negative reciprocals of each other, their product will be \(-1\).

Hence, it can be said that if the slope of two lines is the same, they are known as parallel lines, while if the slope of two given lines is negative reciprocal of each other, they are called perpendicular lines.

Parallel and Perpendicular Lines – Example 1:

If one line passes through the points \((0, –4)\) and \((–1, –7)\) and another line passes through the points \((3, 0)\) and \((–3, 2)\). Are these lines parallel or perpendicular?

Solution:

First, find the slope of each line: \(\color{blue}{slope=\frac{y_2-y_1}{x_2-x_1}}\)

\(m_1 = \frac{(-7-(-4))}{(-1-0)}=\frac{-3}{-1}= 3\)

\(m_2 = \frac{(2-0)}{(-3-3)}= \frac{2}{-6}= \frac{-1}{3}\)

Since, \(m_1 ≠ m_2\), therefore, lines are not parallel.

\(m_1×m_2 = 3×(-\frac{1}{3})= -1\)

Therefore, the two lines are perpendicular.

Exercises for Parallel and Perpendicular Lines

  1. What is the equation for the line that is perpendicular to \(4x−3y=6\) through the point \((4,6)\)?
  2. Find the equation of the line that passes through the point \(A=(-1,2)\) and is parallel to the line \(y=2x-3\).
  3. Find the equation of the line that passes through the point \(A=(1,-2)\) and is perpendicular to the line \(y=3x+4\).
  4. Write the equation for the line that is parallel to \(y=-\frac{1}{3}x+8\) through the point \((0,-2)\).
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{y=-\frac{3}{4}x+9}\)
  2. \(\color{blue}{y=2x+4}\)
  3. \(\color{blue}{y=-\frac{1}{3}x-\frac{5}{3}}\)
  4. \(\color{blue}{y=-\frac{1}{3}x-2}\)

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