How do you find the value of a given the points (-5,3), (-3,a) with a distance of sqrt5?

Feb 3, 2017

$a = 2$ or $a = 4$

Explanation:

The distance between two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ is given by:

color(magenta)(d = sqrt((y_2 - y_1)^2 + (x_2 - x_1)^2)

Therefore:

$\sqrt{5} = \sqrt{{\left(a - 3\right)}^{2} + {\left(- 3 - \left(- 5\right)\right)}^{2}}$

Square both sides:

${\left(\sqrt{5}\right)}^{2} = {\left(\sqrt{{\left(a - 3\right)}^{2} + 4}\right)}^{2}$

$5 = {\left(a - 3\right)}^{2} + 4$

$5 - 4 = {\left(a - 3\right)}^{2}$

$1 = {\left(a - 3\right)}^{2}$

Hence either $a - 3 = 1$ i.e. $a = 4$

or $a - 3 = - 1$ i.e. $a = 2$

Hopefully this helps!