Question

How do you find the distance between $\left( 7,2 \right),\left( -5,7 \right)$ ?

Answer 1

We are asked to find the distance between two given coordinates that lie on the real plane. For this, we use the distance formula. We substitute the values of x – coordinates and the y – coordinates and then evaluate to get the result, which is a positive value, which will be the distance between these two coordinates.

Complete step by step solution:
The given two coordinates are, $\left( 7,2 \right),\left( -5,7 \right)$
We are asked to find the distance between these two points.
For this, we use the distance formula.
First, let us denote some variables for the x – coordinates and the y – coordinates of these points for easy evaluation without confusion.
Let ${{x}_{1}}$ be the x – coordinate of the first point which is 7.
${{x}_{2}}$ will then be the x – coordinate of the second point which is -5.
Let ${{y}_{1}}$ be the x – coordinate of the first point which is 2.
${{y}_{2}}$ will then be the x – coordinate of the second point which is 7.
The distance formula for any two points on the real cartesian plane is given by,
$D=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$

Now let us substitute the values of the x – coordinates and the y – coordinates of these points.
Upon substitution we get,
$\Rightarrow D=\sqrt{{{\left( -5-7 \right)}^{2}}+{{\left( 7-2 \right)}^{2}}}$
Now evaluate the contents inside the brackets first and then the power on it, according to the BODMAS rule.
$\Rightarrow D=\sqrt{{{\left( -12 \right)}^{2}}+{{\left( 5 \right)}^{2}}}$
Further, evaluate the expression.
$\Rightarrow D=\sqrt{144+25}$
$\Rightarrow D=\sqrt{169}$
Now further simplify the square root.
$\Rightarrow D=13$

Hence, the distance between the points is given by the value, 13.

Note: The formula for finding the distance between any two coordinates $({{x}_{1}},{{y}_{1}}),({{x}_{2}},{{y}_{2}})$ is given by, $d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}$ . One should always note that the value resulted will never be negative because it is how far is one point from another and it is always positive.