# What is the distance between (4 ,( - 3pi)/4 ) and (7 ,( 3 pi )/4 )?

Feb 26, 2016

$8.06$

#### Explanation:

Point (4,−(3pi)/4) in Cartesian coordinates represents (4cos(−(3pi)/4)), 4sin(−(3pi)/4))

i.e. $\left(4 \times \left(\frac{- 1}{\sqrt{2}}\right) , 4 \times \left(\frac{- 1}{\sqrt{2}}\right)\right)$ or $\left(- \frac{4}{\sqrt{2}} , - \frac{4}{\sqrt{2}}\right)$ or
$\left(- 2 \sqrt{2} , - 2 \sqrt{2}\right)$

Point $\left(7 , \frac{3 \pi}{4}\right)$ in Cartesian coordinates represents (7cos(3pi/4)),7sin(3pi/4)))

i.e. $\left(7 \times \left(\frac{- 1}{\sqrt{2}}\right) , 7 \times \left(\frac{1}{\sqrt{2}}\right)\right)$ or $\left(- \frac{7}{\sqrt{2}} , \frac{7}{\sqrt{2}}\right)$ or
$\left(- 3.5 \sqrt{2} , 3.5 \sqrt{2}\right)$

Hence distance between $\left(- 3.5 \sqrt{2} , 3.5 \sqrt{2}\right)$ and $\left(- 2 \sqrt{2} , - 2 \sqrt{2}\right)$

is sqrt((-2sqrt2+3.5sqrt2)^2+(-2sqrt2-3.5sqrt2)^2

= sqrt((1.5sqrt2)^2+(-5.5sqrt2)^2

= $\sqrt{2.25 \times 2 + 30.25 \times 2}$

= $\sqrt{4.5 + 60.5}$ = sqrt65)= 8.06