How do you find the value of a given the points (8,-5), (a,4) with a distance of sqrt85?

Nov 14, 2017

See a 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 and solveing for $a$ gives:

$\sqrt{85} = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{8}\right)}^{2} + {\left(\textcolor{red}{4} - \textcolor{b l u e}{- 5}\right)}^{2}}$

$\sqrt{85} = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{8}\right)}^{2} + {\left(\textcolor{red}{4} + \textcolor{b l u e}{5}\right)}^{2}}$

$\sqrt{85} = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{8}\right)}^{2} + {9}^{2}}$

$\sqrt{85} = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{8}\right)}^{2} + 81}$

${\left(\sqrt{85}\right)}^{2} = {\left(\sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{8}\right)}^{2} + 81}\right)}^{2}$

$85 = {\left(\textcolor{red}{a} - \textcolor{b l u e}{8}\right)}^{2} + 81$

$85 = {a}^{2} - 16 a + 64 + 81$

$85 - \textcolor{red}{85} = {a}^{2} - 16 a + 64 + 81 - \textcolor{red}{85}$

$0 = {a}^{2} - 16 a + 60$

${a}^{2} - 16 a + 60 = 0$

$\left(a - 10\right) \left(a - 6\right) = 0$

Solution 1:

$a - 10 = 0$

$a - 10 + \textcolor{red}{10} = 0 + \textcolor{red}{10}$

$a - 0 = 10$

$a = 10$

Solution 2:

$a - 6 = 0$

$a - 6 + \textcolor{red}{6} = 0 + \textcolor{red}{6}$

$a - 0 = 6$

$a = 6$

$a$ can be either $6$ or $10$