Question 179715
Find the distance between these two points: 
(6,4) and (-3,5) 
<pre><font size = 4 color = "indigo"><b>
There are two ways to do this:

Plot the points and connect them with a line segment:

{{{drawing(400,291.1,-4,7,-1,7, grid(1), line(6,4,-3,5) )}}}

Then we draw in these two line segments forming a right
triangle:

{{{drawing(400,291.1,-4,7,-1,7, grid(1), line(6,4,-3,5),
line(-3,4,-3,5), line(-3,4,6,4) )}}}

We count how many units long the bottom (horizontal) side of 
that right triangle is and see that it is 9 units long. Then 
we see that the left (vertical) side of the right triangle 
is only 1 unit long. And the hypotenuse is the distance
between the desired points.  

{{{drawing(400,291.1,-4,7,-1,7, line(6,4,-3,5),
line(-3,4,-3,5), line(-3,4,6,4),locate(1,4,"a=9"),
locate(-4,4.5,"b=1"), locate(1,5,"c=?") )}}}




So we use the Pythagorean 
theorem 

{{{c^2=a^2+b^2}}}
{{{c^2=9^2+1^2}}}
{{{c^2=81+1}}}
{{{c^2=82}}}
{{{sqrt(c^2)=sqrt(82)}}}
{{{c=sqrt(82)}}}

Another way is to memorize the formula

{{{d=sqrt((x[2]-x[1])^2+(y[2]-y[1])^2)}}}

and plug in {{{x[1]=6}}}, {{{x[2]=-3}}}, {{{y[1]=4}}}, {{{y[2]=5}}}

{{{d=sqrt(((-3)-(6))^2+((5)-(4))^2 )}}}

{{{d=sqrt((-9)^2+(1)^2))  }}}

{{{d=sqrt(81+1)}}}

{{{d=sqrt(82)}}}

Either way we get the same answer.

The distance is {{{sqrt(82)}}}

Edwin</pre>