Question

How do you find the distance between $\left( 3,-2 \right)$ and $\left( 5,-3 \right)$?

Answer 1

Hint: To start with, we are to find the distance between $\left( 3,-2 \right)$ and $\left( 5,-3 \right)$. We are going to use the Euclidean distance formula in our problem to get the solution. Putting the right values and simplifying the solution will give us the right result. Applying the Euclidean distance formula is the only needed trick here.

Complete step by step solution:
According to the question, we are to find the distance between $\left( 3,-2 \right)$ and $\left( 5,-3 \right)$.
The Euclidean distance d between two given points $\left( a,b \right)$ and $\left( c,d \right)$is given by the distance formula: $d=\sqrt{{{\left( c-a \right)}^{2}}+{{\left( d-b \right)}^{2}}}$.
In our case, we are given, c = 5, d = -3, a = 3 and b = -2.
Thus, we have the distance as, $d=\sqrt{{{\left( 5-3 \right)}^{2}}+{{\left( \left( -3 \right)-\left( -2 \right) \right)}^{2}}}$
Now, simplifying we get, $d=\sqrt{{{\left( 5-3 \right)}^{2}}+{{\left( -3+2 \right)}^{2}}}$
Again, subtracting the numbers and writing them down, we will get,
$\Rightarrow d=\sqrt{{{\left( 2 \right)}^{2}}+{{\left( -1 \right)}^{2}}}$
Removing the squares now, we are getting,
$\Rightarrow d=\sqrt{4+1}=\sqrt{5}$units.
So, the distance of the two given points is, $\sqrt{5}$ units.
Thus, the solution is $\sqrt{5}$ units.

Note: Sometimes you need to find the point that is exactly between two other points. This middle point is called the "midpoint". By definition, a midpoint of a line segment is the point on that line segment that divides the segment in two congruent segments. In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance.