# How do you find a possible value for a if the points (-9,-2), (a, 5) has a distance of d=7?

Mar 18, 2017

See the entire solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

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

Now, substitute $d$ and the values from the points given in the problem and solve for $a$:

$7 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{- 9}\right)}^{2} + {\left(\textcolor{red}{5} - \textcolor{b l u e}{- 2}\right)}^{2}}$

$7 = \sqrt{{\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + {\left(\textcolor{red}{5} + \textcolor{b l u e}{2}\right)}^{2}}$

$7 = \sqrt{{\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + {\left(7\right)}^{2}}$

$7 = \sqrt{{\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + 49}$

${7}^{2} = {\left(\sqrt{{\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + 49}\right)}^{2}$

$49 = {\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + 49$

$49 - \textcolor{red}{49} = {\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + 49 - \textcolor{red}{49}$

$0 = {\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2} + 0$

$0 = {\left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)}^{2}$

$0 = \left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right) \left(\textcolor{red}{a} + \textcolor{b l u e}{9}\right)$

We can now solve $a + 9$ for $0$:

$a + 9 = 0$

$a + 9 - \textcolor{red}{9} = 0 - \textcolor{red}{9}$

$a + 0 = - 9$

$a = - 9$

$- 9$ is a possible value for $a$.