Question

# How do you find the distance between the points (4,2), (6,-2/3)?

How do you find the distance between the points (4,2), (6,-2/3)?

See the entire solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

$d = \sqrt{{\left(\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}\right)}^{2} + {\left(\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}\right)}^{2}}$

Substituting the values from the points in the problem gives:

$d = \sqrt{{\left(\textcolor{red}{6} - \textcolor{b l u e}{4}\right)}^{2} + {\left(\textcolor{red}{- \frac{2}{3}} - \textcolor{b l u e}{2}\right)}^{2}}$

$d = \sqrt{{2}^{2} + {\left(\textcolor{red}{- \frac{2}{3}} - \left(\frac{3}{3} \times \textcolor{b l u e}{2}\right)\right)}^{2}}$

$d = \sqrt{4 + {\left(\textcolor{red}{- \frac{2}{3}} - \frac{6}{3}\right)}^{2}}$

$d = \sqrt{4 + {\left(- \frac{8}{3}\right)}^{2}}$

$d = \sqrt{4 + \frac{64}{9}}$

$d = \sqrt{\left(\frac{9}{9} \times 4\right) + \frac{64}{9}}$

$d = \sqrt{\frac{36}{9} + \frac{64}{9}}$

$d = \sqrt{\frac{36 + 64}{9}}$

$d = \sqrt{\frac{100}{9}}$

$d = \frac{\sqrt{100}}{\sqrt{9}}$

$d = \frac{10}{3}$

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