College

Using the slope formula, determine if lines [tex]L_1[/tex] and [tex]L_2[/tex] are parallel, perpendicular, or neither.

Given points:
- [tex]L_1[/tex]: (6, 2) and (8, -2)
- [tex]L_2[/tex]: (5, 1) and (3, 0)

Answer :

The lines L₁ and L₂ are perpendicular to each other.

To determine whether the lines L₁ and L₂ are parallel, perpendicular, or neither, we can use the slope formula.

Find the slope of line L₁ using the points (6, 2) and (8, -2). The slope (m₁) is calculated using the formula:

[tex]m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 2}{8 - 6} = \frac{-4}{2} = -2[/tex]

So, the slope of L₁ is -2.

Find the slope of line L₂ using the points (5, 1) and (3, 0). The slope (m₂) is calculated using the same formula:

[tex]m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{3 - 5} = \frac{-1}{-2} = \frac{1}{2}[/tex]

So, the slope of L₂ is [tex]\frac{1}{2}[/tex].

Compare the slopes:

Lines are parallel if their slopes are equal: [tex]m_1 = m_2[/tex]

Lines are perpendicular if the product of their slopes equals -1: [tex]m_1 m_2 = -1[/tex]

Calculate the product of the slopes:

[tex]m_1 m_2 = -2 \times \frac{1}{2} = -1[/tex]

Since [tex]m_1 m_2 = -1[/tex], this indicates that the lines L₁ and L₂ are perpendicular.