Question

Find the distance from P to l. Line l contains points (6,5) and (2,3). Point P has coordinates (2,6).

Answer 1

ok so the way to do this is like this


The brute-force way to do this is:

Find the slope of Line_I :

  Mɪ = (5 – 3) ⁄ (6 – 2) = +½

The equation of Line_I is :

     y = mx + b

     y = (½)x + b ... substitute either point on Line_I

     5 = (½)(6) + b

     b = 2

     y = (½)x + 2 ... Line_I equation


The slope of the perpendicular (shortest) line from
point_P to Line_I = Mpɪ  =  -1 ⁄ (Mɪ) = - 2  ——>  for Line_PI


The equation of Line_PI is :
 y = mx + b

 y = - 2x + b ... substitute point_P

 6 = - 2(2) + b

 b = 10

 y = - 2x + 10 ... Line_PI equation


The intersection between Line_I and Line_PI is located at :
        y = y
    (½)x + 2 = - 2x + 10
        x = 3.2

and the y_value at that x_location is:
  y = (½)x + 2 ... Line_I equation
  y = (½)(3.2) + 2
  y = 3.6

       y = - 2x + 10 ... Line_PI equation
       y = - 2(3.2) + 10
       y = 3.6  ...  checks out


 ... so the intersection is at point_X = (3.2, 3.6)

 ... and the distance from point_P to point_X is :

   d = √  [ (∆x)² + (∆y)²  ]

   d = √  [  (3.2 – 2)² + (6 – 3.6)²   ]

   d = √  7.2  =  √  (36 ⁄ 5)  =  6(√ 5) ⁄ 5  =  2.68

Answer 2

does this make since?

Answer 3

Yesss. Thank you !!!!!