# How do you find the value of a given the points (1,1), (a,1) with a distance of 4?

Mar 14, 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 from the problem gives and solving for $a$ gives:

$4 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{1} - \textcolor{b l u e}{1}\right)}^{2}}$

$4 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{1}\right)}^{2} + {0}^{2}}$

$4 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{1}\right)}^{2} + 0}$

4 = sqrt((color(red)(a) - color(blue)(1))^2

$4 = \textcolor{red}{a} - \textcolor{b l u e}{1}$ and $4 = - \left(\textcolor{red}{a} - \textcolor{b l u e}{1}\right)$ (Note: The square root of a term always produces a negative and positive result)

Solution 1)

$4 = a - 1$

$4 + \textcolor{red}{1} = a - 1 + \textcolor{red}{1}$

$5 = a - 0$

$5 = a$

$a = 5$

Solution 2)

$4 = - \left(a - 1\right)$

$4 = - a + 1$

$4 - \textcolor{red}{1} = - a + 1 - \textcolor{red}{1}$

$3 = - a + 0$

$3 = - a$

$\textcolor{red}{- 1} \times 3 = \textcolor{red}{- 1} \times - a$

$- 3 = a$

$a = - 3$

Solution: $a$ can be either $5$ or $- 3$