# What is the distance between (3 , (3 pi)/4 ) and (9, pi )?

Mar 14, 2016

$14.739$

#### Explanation:

$\left(r , \theta\right)$ in polar coordinates is $\left(r \cos \theta , r \sin \theta\right)$ in rectangular coordinates.

Hence $\left(3 , \frac{3 \pi}{4}\right)$ is $\left(3 \cos \left(\frac{3 \pi}{4}\right) , 3 \sin \left(\frac{3 \pi}{4}\right)\right)$ or $\left(3 \cdot \frac{- 1}{\sqrt{2}} , 3 \cdot \frac{- 1}{\sqrt{2}}\right)$ or $\left(\frac{- 3 \sqrt{2}}{2} , - \frac{3 \sqrt{2}}{2}\right)$

and 9.pi) is $\left(9 \cos \pi , 9 \sin \pi\right)$ or $\left(9 \times 1 , 9 \times 0\right)$ or $\left(9 , 0\right)$

The distance between $\left(9 , 0\right)$ and $\left(\frac{- 3 \sqrt{2}}{2} , - \frac{3 \sqrt{2}}{2}\right)$ is

$\sqrt{{\left(9 - \left(\frac{- 3 \sqrt{2}}{2}\right)\right)}^{2} + {\left(0 - \left(\frac{- 3 \sqrt{2}}{2}\right)\right)}^{2}}$ or

$\sqrt{{\left(9 + \frac{3 \sqrt{2}}{2}\right)}^{2} + {\left(\frac{3 \sqrt{2}}{2}\right)}^{2}}$ or

$\sqrt{81 + 27 \sqrt{2} + \frac{9}{2} + \frac{9}{2}}$ or

$\sqrt{90 + 27 \sqrt{2}}$ or $3 \sqrt{10 + 3 \sqrt{2}}$ or

= $3 \sqrt{10 + 14.142} = 3 \sqrt{24.142} = 3 \times 4.913 = 14.739$