# What is the distance between (-9 ,( 17 )/12 ) and (-2 ,(-7 pi )/4 )?

Jan 11, 2016

I got: $d \approx 8.103$ to 3 decimal places

Perhaps some one else can spot an easier way of working it out!

#### Explanation:

$\textcolor{b l u e}{\text{Assumption: We are looking at the shortest distance between the two Cartesian points.}}$
$\textcolor{b l u e}{\text{Which is a strait line.}}$

$\textcolor{m a \ge n t a}{\text{However, the" (-7pi)/4"implies Polar Coordinate system. It is not stated that this is the case!!!}}$

Ant two points on a strait line graph can be viewed as forming a right triangle. Unless it is of the forms x=n" or y=p where n and p are some constant.

So we are looking at $\left(\text{difference in y-axis")/("difference in x-axis}\right)$

That is $\frac{{x}_{2} - {x}_{1}}{{y}_{2} - {y}_{1}}$

Let $\left({x}_{1} , {y}_{1}\right) \to \left(- 9 , \frac{17}{12}\right)$

Let $\left({x}_{2} , {y}_{2}\right) \to \left(- 2 , \frac{- 7 \pi}{4}\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using Pythagoras

Let the distance between the points be d then

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

${d}^{2} = {\left[\textcolor{w h i t e}{\frac{1}{2}} \left(- 2\right) - \left(- 9\right)\right]}^{2} + {\left[\frac{- 7 \pi}{4} - \frac{17}{12}\right]}^{2}$

${d}^{2} = {7}^{2} + {\left(\frac{17 - 21 \pi}{12}\right)}^{2}$

$d = \sqrt{49 + \frac{2398.398}{144}} \textcolor{w h i t e}{\ldots}$ to 3 decimal places

$d \approx 8.103$ to 3 decimal places