Find the value of k if the line through the points (10-k, k) and (k+4,5k+1) is perpendicular to the line through the points (-1.) and (7,6)

k ≈ 1 . 03

Step-by-step explanation: Note that for two lines to be perpendicular, then the product of their slope must be -1 , i. e if M1 is the slope of the first line and M2 is the slope of the second line, since they are perpendicular, M1 x M2 = -1

Finding the slope of the first line using the formula for finding slope

i.e M1 = Y2 - Y1 / X2 - X1

X1 = 10 - K

X2 = K + 4

Y1 = K

Y2 = 5K + 1

Substituting into the formula , we have

M1 = 5K + 1 -K / K + 4 - (10 - K )

     =  4K + 1 / K + 4 - 10 + K

     = 4K + 1 / 2K - 6

Finding the slope of the second line

M2 = Y2 - Y1/ X2 - X1

X1 = - 1.5

X2 = 7

Y1  = -0.5

Y2 = 6

Substituting into the slope formula

M2 = 6 - ( - 0.5) / 7 - ( - 1.5 )

     = 6 + 0.5 / 7 + 1.5

     = 6.5 / 8.5

Recall that for the two lines to be perpendicular, M1 x M2 = -1

⇒ 4k + 1 / 2k -6   x 6.5 / 8.5 = -1

⇒ 6.5 ( 4k + 1)/ 8.5 ( 2k - 6 ) = -1

⇒ 26k + 6.5 / 17k - 51 = -1

cross multiply

26k + 6.5 = -1 ( 17k - 51)

26k + 6.5 = - 17k  + 51

collect the like terms

26k + 17k = 51 - 6.5

43k          = 44.5

    k          = 44.5/43

           k  ≈ 1. 03

you didn't give me the choices, so i will just give you the graph so you can look at your choices and choose the right one.

step-by-step explanation:

points a and b are equidistant from the desired line, so the line you find by using them will go through point r.

using points other than a and b will give the desired line only if they are equidistant from r. if they are randomly located, the perpendicular line you create will not necessarily go through point r, hence will not be a solution to the problem.