# What is the perimeter of a triangle with corners at (3 ,0 ), (5 ,2 ), and (5 ,4 )?

Mar 22, 2016

$2 \sqrt{2} + 2 \sqrt{5} + 2 \approx 9.3$

#### Explanation:

The distance formula tells us that the distance between two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ is:

$\sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

If ${x}_{1} = {x}_{2}$ then this simplifies to $\left\mid {y}_{2} - {y}_{1} \right\mid$

The distance between $\left(3 , 0\right)$ and $\left(5 , 2\right)$ is:

$\sqrt{{\left(5 - 3\right)}^{2} + {\left(2 - 0\right)}^{2}} = \sqrt{{2}^{2} + {2}^{2}} = \sqrt{8} = 2 \sqrt{2}$

The distance between $\left(3 , 0\right)$ and $\left(5 , 4\right)$ is:

$\sqrt{{\left(5 - 3\right)}^{2} + {\left(4 - 0\right)}^{2}} = \sqrt{{2}^{2} + {4}^{2}} = \sqrt{20} = 2 \sqrt{5}$

This distance between $\left(5 , 2\right)$ and $\left(5 , 4\right)$ is:

$\left\mid 4 - 2 \right\mid = 2$

These are the lengths of the three sides, so the perimeter is just the sum:

$2 \sqrt{2} + 2 \sqrt{5} + 2 \approx 9.3$