Question

How do you find the distance between the \left( -3,-5 \right)$$$$\left( -6,-8 \right)?

Answer 1

From the given question we have to find the distance and midpoint between the points $\left( -3,-5 \right)$ and $\left( -6,-8 \right)$. we know that formula of distance of two points and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$.By the above formulas we will get the required solution.
To find the distance we just apply Pythagoras. Think of it this way:
The difference between the x points causes a straight horizontal line, the difference between the y points causes a straight vertical line, so the distance between the two points is the hypotenuse
$\Rightarrow {{d}^{2}}=\Delta {{x}^{2}}+\Delta {{y}^{2}}$
$\Rightarrow d=\sqrt{\Delta {{x}^{2}}+\Delta {{y}^{2}}}$
Here d is the distance between the two points.
$\Delta x$ is the difference between the x points that causes a straight horizontal line.
$\Rightarrow \Delta x={{x}_{2}}-{{x}_{1}}$
$\Delta y$ is the difference between the x points that causes a straight horizontal line.
$\Rightarrow \Delta y={{y}_{2}}-{{y}_{1}}$

Complete step by step solution:
From the question we have two points they are,
$\Rightarrow \left( -3,-5 \right),\left( -6,-8 \right)$
Firstly, we have to find the distance between these points.
We know that formula for the distance between the points and $\left( {{x}_{2}},{{y}_{2}} \right)$ is
$\Rightarrow \sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$
Therefore, here by comparing

& \Rightarrow {{x}_{1}}=-3,\ {{y}_{1}}=-5 \\\ & \Rightarrow {{x}_{2}}=-6,\ {{y}_{2}}=-8 \\\ \end{aligned}$$ Let D be the distance between the points By substituting the values in the above formula, we will get, $$\Rightarrow D=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$$ $$\Rightarrow D=\sqrt{{{\left( -6-\left( -3 \right) \right)}^{2}}+{{\left( -8-\left( -5 \right) \right)}^{2}}}$$ $$\Rightarrow D=\sqrt{{{\left( -3 \right)}^{2}}+{{\left( -3 \right)}^{2}}}$$ $$\Rightarrow D=\sqrt{9+9}$$ $$\Rightarrow D=\sqrt{18}$$ $$18$$can be written as product of $$9$$and $$2$$ $$\Rightarrow D=\sqrt{9.2}$$ Therefore $$9$$ is the square of $$3$$ $$\Rightarrow D=3\sqrt{2}$$ Therefore, the distance between the two points is $$ D=3\sqrt{2}$$ **Note:** Students should be very careful while doing the calculation like, $$ D=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$$ here in this formula there itself negative signs are there students should substitute the exact values of the points with their signs, if we write $$3$$ in place of $${{x}_{1}}$$ then answer will be changed we should have to write $$-3$$ only.