Question

Find the distance between the following pairs of points:
(2, 3), (4, 1).

Answer 1

Hint: For solving this problem, we consider two points A and B. The coordinates of A are (2, 3) and the coordinates of B are (4, 1). By applying the distance formula between two points we can easily obtain the length AB which is the distance between points.

Complete step-by-step solution -
The distance between any two points in the plane is the length of line segment joining them. Consider two-point P and Q in the xy plane. Let the coordinates of P be $\left( {{x}_{1}},{{y}_{1}} \right)$ and coordinates of Q be $\left( {{x}_{2}},{{y}_{2}} \right)$. The distance between P and Q is given by the formula: $PQ=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$.

According to the problem statement, we are given two points A and B whose coordinates are (2, 3) and (4, 1) respectively. The length of AB which is the distance between the points can be specified by using the above stated formula:
$\begin{align}
  & AB=\sqrt{{{\left( 2-4 \right)}^{2}}+{{\left( 3-1 \right)}^{2}}} \\
 & AB=\sqrt{{{\left( -2 \right)}^{2}}+{{2}^{2}}} \\
 & AB=\sqrt{4+4} \\
 & AB=\sqrt{8} \\
 & AB=2\sqrt{2} \\
\end{align}$
Therefore, the distance between (2, 3) and (4, 1) is $2\sqrt{2}$ units.

Note: This problem can be alternatively solved by using the diagram given below.

 We can form a triangle ABC which is a right angle triangle. By observation, AC and BC are determined. Now, by applying Pythagoras theorem in triangle ABC, we can easily obtain the length AB which is the same as obtained above.