Karlee has been spending the day exploring the woods near her house. It was getting late and Karlee needed to get back home. The quickest way home for Karlee is to follow the bike path on the other side of the woods, In her coordinate map, Karlee determined that her position was (-4,1) and that the bike was located along the line -x-2y+4=0. Determine the shortest distance that Karlee needs to travel to reach the bike path from her current position.

There is a formula for this. 
Given a line ax+by+c=0 and a point, the shortest distance between them is: 

What is Ax+By=Cz

Ax+By+Cz=D is the general form for the equation of a plane, so Ax+By=Cz is a plane 


:)

If the point (a, 3) is at a distance of √5 unit from (2, a), find a?

 
which leads is to a=1 or a=4 



:)

A line segment has a midpoint of (–3, 2) and an endpoint of (–6, 4). What is the other endpoint of the line segment?

The midpoint coordinates are: 

 


 

Substituting... 




 


 

After simplifying we get (0,0) 

:)

find the distance from (6,-4) to the line defined by y=2x-6

Typically we ask for the shortest distance because the distance from the line varies. So here we go... 

There is a line perpendicular to our line that passes through (6,-4). Since it is perpendicular it has the slope -1/2 

 


 

After some simplifying we can get 

 



Notice that this new line must intersect our original at a point exactly parallel to (6,-4) (why?) 

Therefore if we find the point of intersection we can use that point to find our distance 


 
and this leads us to the x-coordinate which is x=2 and so y=2(2)-6=-2 

Next we find the distance between (6,-4) and (2,-2) 


 
and this gives us 

 



Actually in precalculus there is a formula: 
 



:)