# What is the perimeter of a triangle with corners at (3 ,8), (1 ,5 ), and (4 ,3 )?

Sep 30, 2016

Perimeter:$\textcolor{m a \ge n t a}{\sqrt{13} \left(2 + \sqrt{2}\right)}$

#### Explanation:

Distance from color(red)(""(3,8)) to color(blue)(""(1,5))
$\textcolor{w h i t e}{\text{XXX}} {d}_{1} = \sqrt{{\left(\textcolor{red}{3} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{8} - \textcolor{b l u e}{5}\right)}^{2}} = \sqrt{4 + 9} = \textcolor{m a \ge n t a}{\sqrt{13}}$

Distance from color(blue)(""(1,5)) to color(green)(""(4,3))
$\textcolor{w h i t e}{\text{XXX}} {d}_{2} = \sqrt{{\left(\textcolor{b l u e}{1} - \textcolor{g r e e n}{4}\right)}^{2} + {\left(\textcolor{b l u e}{5} - \textcolor{g r e e n}{3}\right)}^{2}} = \sqrt{9 + 4} = \textcolor{m a \ge n t a}{\sqrt{13}}$

Distance from color(green)(""(4,3))to color(red)(""(3,8))
color(white)("XXX")d_3=sqrt((color(green)4-color(red)3)^2+(color(green)3-color(red)8)^2)=sqrt(1+25)=color(magenta)(sqrt(26)=sqrt(2) * sqrt(13)

color(magenta)("Perimeter" = d_1+d_2+d_3

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{m a \ge n t a}{2 \sqrt{13} + \sqrt{2} \sqrt{13}}$

$\textcolor{w h i t e}{\text{XXX}} = \textcolor{m a \ge n t a}{\sqrt{13} \left(2 + \sqrt{2}\right)}$