Determine whether the lines L1 and L2 passing through the pairs of points are parallel, perpendicular, or neither.
L1: (2, −3), (4, 3)
L2: (3, 8), (9, 6)
parallel
perpendicular
neither

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?

---

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 \(-1\).

Learn how to determine if two lines passing through given points are parallel, perpendicular, or neither using slope calculations. Practice with examples and understand the conditions for parallel and perpendicular lines in geometry. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation