# How do you find the value of a given the points (2,a), (2,3) with a distance of 10?

Apr 13, 2017

See the entire solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

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

Substituting the values given in the problem for the distance and the points gives:

$10 = \sqrt{{\left(\textcolor{red}{2} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)}^{2}}$

We can now solve for $a$:

$10 = \sqrt{{0}^{2} + {\left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)}^{2}}$

$10 = \sqrt{0 + {\left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)}^{2}}$

$10 = \sqrt{{\left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)}^{2}}$

Remember, taking the square root results in a positive and negative result:

$10 = \pm \left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)$

Solution 1)

$10 = + \left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)$

$10 = \textcolor{red}{3} - \textcolor{b l u e}{a}$

$- 3 + 10 = - 3 + \textcolor{red}{3} - \textcolor{b l u e}{a}$

$7 = 0 - \textcolor{b l u e}{a}$

$7 = - \textcolor{b l u e}{a}$

$- 1 \cdot 7 = - 1 \cdot - \textcolor{b l u e}{a}$

$- 7 = a$

$a = - 7$

Solution 2)

$10 = - \left(\textcolor{red}{3} - \textcolor{b l u e}{a}\right)$

$10 = - \textcolor{red}{3} + \textcolor{b l u e}{a}$

$3 + 10 = 3 - \textcolor{red}{3} + \textcolor{b l u e}{a}$

$13 = 0 + \textcolor{b l u e}{a}$

$13 = \textcolor{b l u e}{a}$

$a$ can equal either $- 7$ or $13$