Straight line between X(-2, -3) and Y(6, 2). Road connecting that road and Z(1, 2). How long would this road be?

Question

There are two campgrounds located at X(-2, -3) and Y(6, 2).

These two campgrounds are connected by a straight dirt road. The campground officers want to build a patrol cabin at Z(1, 2).

They also need to build a road to connect the patrol cabin to the road between the two campgrounds.

 

How can the builders build the road with using minimal supplies?

 

How long would this road be?

Answer 1

 

The slope of XY is M = (2 - -3)/(6 - -2) = 5/8

Using Y(6,2), the equation of route XY  has intercept B = y - mx = 2 - (5/8)(6) = 2 - 30/8 = 16/8 - 30/8 = -14/8 = -7/4

So the equation of route XY is y = (5/8)X - 7/4

 

The line perpendicular to XY must have slope -8/5 and pass through Z.

 

THerefore, the line perpendicular to route XY has intercept B = y - mX =  2 - (-8/5)(1) = 2 + 8/5 = 10/5+8/5 = 18/5

So the line perpendicular to route XY is y = (-8/5)x + 18/5

 

They intersect at :

 (5/8)x - 7/4  = (-8/5)x + 18/5

 

Multiplying everything by common denominator 40:

 

  25x - 70 = -64X + 144

 

  25 x - 70 + 64x - 144 = 0 <---- everything to left side

 

 89x  - 214 = 0

 

 x = 214/89

 

THen y = (5/8)(214/89) - 7/4

            = 1070/712 - 7/4 

            = 1070/712 -  1246/712  <---  common denominator

           = -176/712

           = -88/356

           = -44/178

           = -22/89

 

 

 

       and

 

     y  = (-8/5)(214/89) + 18/5

           = -1712/445 + 18/5

            = -1712/445 +  1602/445

             = -110/445

             = -22/89 

 

 

So the road from the patrol tower at Z to the main route XY intersect at (214/89, -22/89)

 

THe distance from the patrol tower at Z to the main route XY is

 

  the distance between Z(1,2) and ( 214/89, -22/89)

 

   OR

 

    sqrt (   (2 - (-22/89))^2  +  (1 - 214/89)^2 ) =

 

    sqrt (   ( 178+ 22)/89 )^2 +   (( 89 - 214)/89))^2 ) =

 

    sqrt (   (200/89)^2 +  (  -125/89)^2 ) =

 

   1/89 * sqrt (  200^2 + (-125)^2 )  <---- 1/89 factors out of the denominator

 

  1/89 * sqrt ( 40000 +    15625) =

 

  1/89 * sqrt ( 55625) =

 

   1/89 * sqrt ( 125 * 445 ) =

 

  1/89 * sqrt ( 5 * 5 *5 * 5 *89) =

 

  1/89 * 25 * sqrt(89)

 

  or approximately = 2.65