# What is the distance between points( 6 , 9 ) and( 6 , − 9 ) on a coordinate plane?

Jun 21, 2018

$18$

#### Explanation:

Given two points ${P}_{1} = \left({x}_{1} , {y}_{1}\right)$ and ${P}_{2} = \left({x}_{2} , {y}_{2}\right)$, you have four possibilities:

• ${P}_{1} = {P}_{2}$. In this case, the distance is obviously $0$.

• ${x}_{1} = {x}_{2}$, but ${y}_{1} \setminus \ne {y}_{2}$. In this case, the two points are vertically aligned, and their distance is the difference between the $y$ coordinates: $d = | {y}_{1} - {y}_{2} |$.

• ${y}_{1} = {y}_{2}$, but ${x}_{1} \setminus \ne {x}_{2}$. In this case, the two points are horizontally aligned, and their distance is the difference between the $x$ coordinates: $d = | {x}_{1} - {x}_{2} |$.

• ${x}_{1} \setminus \ne {x}_{2}$ and ${y}_{1} \setminus \ne {y}_{2}$. In this case, the segment connecting ${P}_{1}$ and ${P}_{2}$ is the hypotenuse of a right triangle whose legs are the difference between the $x$ and $y$ coordinates, so by Pythagoras we have

$d = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

Note that this last formula covers all the previous cases as well, although it is not the most immediate.

So, in your case, we can use the second bullet point to compute

$d = | 9 - \left(- 9\right) | = | 9 + 9 | = 18$