Answer:
k = - 7.5
Step-by-step explanation:
Slope "m" of a line through points
( [tex]x_{1}[/tex] , [tex]y_{1}[/tex] )
( [tex]x_{2}[/tex] , [tex]y_{2}[/tex] )
is m = [tex]\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]
Slopes of perpendicular lines are opposite reciprocal: [tex]m_{1}[/tex] [tex]m_{2}[/tex] = - 1
~~~~~~~~~~~~~~~
The slope of a line through points (0, 7) and (2, 10)
[tex]m_{1}[/tex] = [tex]\frac{10-7}{2-0}[/tex] = [tex]\frac{3}{2}[/tex]
Slope of perpendicular line through points (3, 5) and (k, 12) is [tex]m_{2}[/tex] = [tex]-\frac{2}{3}[/tex]
or [tex]m_{2}[/tex] = [tex]\frac{12-5}{k-3}[/tex]
[tex]\frac{7}{k-3}[/tex] = [tex]-\frac{2}{3}[/tex]
2(k - 3) = - 21
k - 3 = - 10.5
k = - 7.5
