Answer: 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