Question

How do you find the length of the line segment between the points $\left( {5,1} \right)$ and $\left( {5,6} \right)$?

Answer 1

For solving this question, we need to use the distance formula in terms of the coordinates of two points. The coordinates of the endpoints of the line segment are given in the question. So by putting these coordinates into the distance formula, we will get the distance between the endpoints of the given line segment. Since the line segment passes through these points, its length will be equal to the distance between the endpoints.

Formula used:
The formula used to solve this question is given by
$d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$, here $d$ is the distance between two points in a plane having the coordinates $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$.

Complete step-by-step solution:
Let us label the given line segment with the coordinates as AB with the respective coordinates A$\left( {5,1} \right)$ and B$\left( {5,6} \right)$.
Now, the length of the line segment will be equal to the distance between the endpoints. So the length of the line segment will be equal to the distance between the points A and B. We know from the distance formula that
$d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$
So the distance AB is given as
$AB = \sqrt {{{\left( {5 - 5} \right)}^2} + {{\left( {6 - 1} \right)}^2}}$
$\Rightarrow AB = \sqrt {{0^2} + {5^2}}$
On solving, we get
$AB = 5$

Hence, the length of the given line segment is equal to $5$ units.

Note:
We could attempt this question without using the distance formula as well. If we carefully note the coordinates of the endpoints of the given line segment, then we will notice that the x-coordinates of both the points are equal. This means that the given line segment is vertical. So the length of the line segment must be equal to the magnitude of the difference between the y-coordinates of the endpoints.