Question

The distance between $\left( 2,3 \right)$ and $\left( -4,5 \right)$ is _____

Answer 1

For solving this problem, we consider two points A and B. The coordinates of A are $\left( 2,3 \right)$ and $\left( -4,5 \right)$. 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 answer:
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 $\left( 2,3 \right)$ and $\left( -4,5 \right)$ respectively. The length of AB which is the distance between the points can be specified by using the above stated formula:
Here we have ${{x}_{1}}=2$, ${{x}_{2}}=-4$, ${{y}_{2}}=5$, ${{y}_{1}}=3$
$AB=\sqrt{{{\left( -4-2 \right)}^{2}}+{{\left( 5-3 \right)}^{2}}}$
$AB=\sqrt{{{\left( -6 \right)}^{2}}+{{\left( 2 \right)}^{2}}}$
$AB=\sqrt{36+4}$
$AB=\sqrt{40}$
$AB=\sqrt{4\times 10}$
$AB=2\sqrt{10}\approx 6.324$
Therefore, this distance between the $\left( 2,3 \right)$ and $\left( -4,5 \right)$ is $2\sqrt{10}\approx 6.324$ units.

Note: Students will make mistake in substituting the values ${{x}_{1}},{{x}_{2}},{{y}_{1}},{{y}_{2}}$in the formula. They may interchange and substitute. At the time of interchange substitution, we may not get the correct solution. In this case we have to remember the formula for distance. We should always remember to find the length between any two given points, we can use the distance formula $PQ=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$.