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Parallel and perpendicular lines

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Explainer Video

Tutor: Daniel

Summary

Parallel and perpendicular lines

In a nutshell

Lines can be parallel or perpendicular to each other. You can find out by looking at the gradients of the lines. If two lines are parallel, they share the same gradient. Two perpendicular lines will have the product of their gradients equal to −1-1−1. If you have the gradient of one line (m)(m)(m), the gradient of a perpendicular line is equal to −1m-\dfrac{1}{m}−m1​.



Parallel lines

Two lines that are parallel will have the same gradient. This is apparent if you look at a graph with parallel lines. They share the same value of mmm.


Maths; Straight line graphs; KS5 Year 12; Parallel and perpendicular lines


Example 1

Prove that the lines l1 (y=3x+2)l_1\space (y=3x+2)l1​ (y=3x+2)​ and l2 (6x−2y−6=0)l_2\space (6x-2y -6=0)l2​ (6x−2y−6=0)​ are parallel.

​

Rearrange equation to give y=mx+cy=mx+cy=mx+c . See if mmm of l1l_1l1​ is equal to mmm of l2l_2l2​.


​​

​l1 m=3l2 is 6x−2y−6=0−2y=−6x+62y=6x−6y=3x−3l2 m=3\begin{aligned}l_1 \space m&=3\\l_2 \space is \space 6x-2y-6&=0\\-2y&=-6x+6\\2y&=6x-6\\y&=3x-3\\l_2\space m&=3\\ \end{aligned}l1​ ml2​ is 6x−2y−6−2y2yyl2​ m​=3=0=−6x+6=6x−6=3x−3=3​

​​​

​l1=l2=3‾\underline{l_1= l_2=3}l1​=l2​=3​, therefore the lines are parallel.



Perpendicular lines

For perpendicular lines, the slope of one line is the negative reciprocal of its perpendicular counterpart. Take the gradient of one line and find the negative reciprocal (−1m)\Big (-\dfrac{1}{m}\Big )(−m1​) to find the gradient of the perpendicular line. The product of the gradients of the two perpendicular lines will equal −1-1−1, m1×m2=−1m_1\times m_2=-1m1​×m2​=−1.


Example 2

Investigate if the lines y=4x+2y=4x +2y=4x+2 and 2x+8y−12=02x+8y-12=02x+8y−12=0 are perpendicular.


Rearrange equation to give y=mx+cy=mx+cy=mx+c.

2x+8y−12=08y=−2x+12y=−28x+128y=−14x+32m=−14\begin{aligned}2x+8y-12&=0\\8y&=-2x+12\\y&=-\dfrac28x+\dfrac{12}{8}\\\\y&=-\dfrac14x+\dfrac32\\\\m&=-\dfrac14\\\end{aligned}2x+8y−128yyym​=0=−2x+12=−82​x+812​=−41​x+23​=−41​​​​

Multiply the gradients to get −1-1−1.


​m1=4,m2=−14m_1 = 4, m_2=-\dfrac14m1​=4,m2​=−41​

4×−14=−14\times -\dfrac14=-14×−41​=−1


m1×m2=−1‾\underline{m_1 \times m_2=-1}m1​×m2​=−1​, therefore the lines are perpendicular.

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FAQs - Frequently Asked Questions

How do you find the gradient of a line L2, which is perpendicular to another line L1?

The value of m for L1 multiplied by the value of m for L2 will be equal to -1 if the lines are perpendicular. The gradient of L2 is equal to -1÷(gradient of L1).

How do you know if two lines are perpendicular?

Two perpendicular lines will have the product of their gradients equal to -1.

How do you know if two lines are parallel?

If two lines are parallel, they share the same gradient.

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