Perpendicular Lines Formula
The perpendicular lines formula is used to find whether two given lines are perpendicular to each other or not. The perpendicular lines formula is applicable when the slope of two lines that we want to compare is known to us. The angle between the two lines which are perpendicular to each other is 90 degrees.
Let us learn more about the formula of the perpendicular line along with solved examples.
What Is Perpendicular Lines Formula?
For any two lines with equations \(y = m_1x+ c_1\) and \(y = m_2x + c_2\), the formula to know that the lines are perpendicular is:
\(m_1 \times m_2 = 1\)
Or, \(m_1 =  \dfrac {1}{m_2}\)
Where,
m_{1} and m_{2} are the slopes of the two lines.
Solved Examples Using Perpendicular Lines Formula

Example 1:
Find out whether the lines 4y + 2x 10 = 0, and y = 2x + 27 are perpendicular to each other or not.
Solution:
To Find: Perpendicular lines
Given:
Equation of line 1: 4y + 2x 10 = 0
On rearranging and dividing by 4, we get
\(y = \dfrac12x + \dfrac52\)
On comparing with \(y = m_1x+ c_1\)
\(\implies m_1 = \dfrac12\)
Now, equation of line 2: y = 2x + 27
On comparing with \(y = m_2x+ c_2\)
\(\implies m_2 = 2\)
Using the perpendicular lines formula,
\(m_1 \times m_2 = 1\)
\(\dfrac12 \times 2 = 1\)
Answer: Hence, the given lines are perpendicular to each other. 
Example 2:
What will be the slope of the line perpendicular to the line 3y  45x = 12?
Solution:
To Find: Slope of the perpendicular line
Given:
Equation of line 1: 3y  45x = 12
On rearranging and dividing by 3, we get: y = 15x + 4
On comparing with \(y = m_1x+ c_1\)
\( \implies m_1 = 15\)
Using the perpendicular lines formula,
\(m_1 \times m_2 = 1\)
\(m_2 =  \dfrac{1}{15}\)
Answer: Hence, the slope of the perpendicular line is \(  \dfrac{1}{15}\).
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