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

answer: ♡♡♡♡♡♡♡♡♡♡

step-by-step explanation: