Question

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

Explanation:

Step1: Find the slope of line KL

Line KL is a horizontal line (parallel to the x - axis). The formula for the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a horizontal line, \(y_1=y_2\). Let \(K(- 8,1)\) and \(L(6,1)\), then \(m_{KL}=\frac{1 - 1}{6-(-8)}=\frac{0}{14} = 0\)

Step2: Find the slope of line MN

Line MN is a vertical line (parallel to the y - axis). For a vertical line, \(x_1=x_2\). Let \(M(-4,8)\) and \(N(-4,-6)\), then the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) results in a division by zero (\(x_2 - x_1=-4-(-4) = 0\)), so the slope of MN is undefined.

Step3: Analyze the relationship between the lines

A line with slope \(m_1 = 0\) (horizontal line) and a line with an undefined slope (vertical line) are perpendicular. The product of the slopes of two perpendicular lines \(m_1\times m_2=- 1\). Here, a horizontal line (slope \(m_1 = 0\)) and a vertical line (undefined slope) satisfy the geometric condition of perpendicularity (they intersect at a right - angle). Also, check the intersection point:

• The lines KL and MN intersect at point \(P(-4,1)\) (not at point K).

Answer:

The lines are perpendicular.