Question

What is the length of AB, given $A\left( 5,-2 \right)$ and $B\left( -3,-4 \right)$?

Answer 1

In this problem we have to find the distance between the given points $A\left( 5,-2 \right)$ and $B\left( -3,-4 \right)$. In this problem, we can use the distance formula to find the length of AB, We can substitute the x and y values in the formula $d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}$, to find the length.

Complete step by step solution:
Here we have to find the length of AB, given $A\left( 5,-2 \right)$ and $B\left( -3,-4 \right)$.
We can now use the distance formula to find the length,
We know that the distance formula is,
$d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}$
We know that the given points can be written as,
$\Rightarrow \left( {{x}_{1}},{{y}_{1}} \right)=\left( 5,-2 \right),\left( {{x}_{2}},{{y}_{2}} \right)=\left( -3,-4 \right)$
We can now substitute the above values and simplify them so we can get the distance from the given point from the origin. By substituting and simplifying them we get,
$\Rightarrow d(AB)=\sqrt{{{\left( -3-5 \right)}^{2}}+{{\left( -4+2 \right)}^{2}}}$
We can now simplify the above step, we will get square root of 100, which will be equal to 10, we get
$\Rightarrow d(AB)=\sqrt{64+36}=\sqrt{100}=10$
Therefore, the distance between the length of AB, given $A\left( 5,-2 \right)$ and $B\left( -3,-4 \right)$.

Note: Students will make mistakes to find the second point where it is given as origin. And they will make mistake in substituting the values ${{x}_{1}},{{y}_{1,}}{{x}_{2}},{{y}_{2}}$ 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 $d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}$.