To determine whether the lines $L_1$ and $L_2$ are parallel, perpendicular, or neither, we need to calculate the slopes of the lines.

Step 1: Find the slope of $L_1$

The formula for the slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For $L_1$, passing through points $(2, -3)$ and $(4, 3)$:

$m_1 = \frac{3 - (-3)}{4 - 2} = \frac{3 + 3}{2} = \frac{6}{2} = 3$

So, the slope of $L_1$ is $m_1 = 3$.

Step 2: Find the slope of $L_2$

For $L_2$, passing through points $(3, 8)$ and $(9, 6)$:

$m_2 = \frac{6 - 8}{9 - 3} = \frac{-2}{6} = -\frac{1}{3}$

So, the slope of $L_2$ is $m_2 = -\frac{1}{3}$.

Step 3: Compare the slopes

• Parallel Lines: If the slopes are equal ($m_1 = m_2$).
• Perpendicular Lines: If the product of the slopes is $-1$ ($m_1 \times m_2 = -1$).
• Neither: If neither condition is met.

Calculate the product of the slopes:

$m_1 \times m_2 = 3 \times \left(-\frac{1}{3}\right) = -1$

Since the product of the slopes is $-1$, the lines $L_1$ and $L_2$ are perpendicular.

Conclusion:

The lines $L_1$ and $L_2$ are perpendicular.

Would you like more details or have any questions?

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Here are 5 related questions for further practice:

1. What are the conditions for two lines to be parallel in a coordinate plane?
2. How would you find the equation of a line perpendicular to $L_1$?
3. If you translate the points of $L_1$ by a certain vector, how does that affect the slope?
4. What is the slope of a line that is parallel to the x-axis?
5. How do you determine the angle between two intersecting lines given their slopes?

Tip: When checking if two lines are perpendicular, always multiply their slopes to see if the result is $-1$.