Question

Find the distance between the pair of points with the given coordinates $\left(\frac{2\sqrt{3}}{3},\frac{\sqrt{5}}{4}\right)\text{and}\left(-\frac{2\sqrt{3}}{3},\frac{\sqrt{5}}{2}\right)$.

Short answer

The distance between the given pair of points is $\frac{\sqrt{813}}{12}\text{units}$.

Step-by-step solution

Step 1. Write down the given information.

The given pair of points have coordinates $\left(\frac{2\sqrt{3}}{3},\frac{\sqrt{5}}{4}\right)\text{and}\left(-\frac{2\sqrt{3}}{3},\frac{\sqrt{5}}{2}\right)$.

Step 2. Concept used.

If a line segment has end-points at $\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)$ then the distance $\left(d\right)$ between these points is given as:

$\left|d\right|=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}....\left(1\right)$

Step 3. Calculation.

The distance $\left(d\right)$between the pair of points with given coordinates $\left(\frac{2\sqrt{3}}{3},\frac{\sqrt{5}}{4}\right)\text{and}\left(-\frac{2\sqrt{3}}{3},\frac{\sqrt{5}}{2}\right)$is evaluated using the formula (1) stated above.

$\begin{array}{c}\left|d\right|=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-\frac{2\sqrt{3}}{3}-\frac{2\sqrt{3}}{3}\right)}^2+{\left(\frac{\sqrt{5}}{2}-\frac{\sqrt{5}}{4}\right)}^2}\\ =\sqrt{{\left(-\frac{4\sqrt{3}}{3}\right)}^2+{\left(\frac{\sqrt{5}}{4}\right)}^2}\\ =\frac{\sqrt{813}}{12}\text{units}\end{array}$

Step 4. Conclusion.

The distance between the given pair of points is $\frac{\sqrt{813}}{12}\text{units}$$\frac{\sqrt{813}}{12}\text{units}$.