Question

how are lines kl and mn related?
the lines intersect at point k.
the lines are parallel.
the lines are perpendicular.
the lines do not have slopes.

Explanation:

Step1: Find the slope of line \(MN\)

Line \(MN\) is a horizontal line. For a horizontal line \(y = c\) (where \(c\) is a constant), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \(M(x_1,y_1)=(8,4)\) and \(N(x_2,y_2)=(- 6,4)\). Then \(m_{MN}=\frac{4 - 4}{8-(-6)}=\frac{0}{14}=0\).

Step2: Find the slope of line \(KL\)

Line \(KL\) is a vertical line. For a vertical line \(x = k\) (where \(k\) is a constant), if we take two points \(K(x_1,y_1)=(2,8)\) and \(L(x_2,y_2)=(2,-6)\), then \(m_{KL}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-6 - 8}{2 - 2}\), and division by zero is undefined. A vertical line and a horizontal line satisfy the condition for perpendicular lines (the product of their slopes is \(m_{MN}\times m_{KL}=0\times\) (undefined) which is equivalent to the geometric property of perpendicularity for horizontal - vertical line pairs).

Answer:

The lines are perpendicular.