If two points are given from each of two lines L1 and L2, without graphing the points, determine if the lines are parallel, perpendicular, or neither: L1: (2,5) and (4,9). L2: (-1,4) and (3,2)

To determine whether two lines are parallel, perpendicular, or neither, we first calculate the slopes of each line using the coordinates of the points on each line. 

The slope of a line is given by the formula:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

### Step 1: Find the slope of Line \( L_1 \)
For \( L_1 \), the points are \( (2,5) \) and \( (4,9) \).

\[
\text{slope of } L_1 = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2
\]

### Step 2: Find the slope of Line \( L_2 \)
For \( L_2 \), the points are \( (-1,4) \) and \( (3,2) \).

\[
\text{slope of } L_2 = \frac{2 - 4}{3 - (-1)} = \frac{-2}{3 + 1} = \frac{-2}{4} = -\frac{1}{2}
\]

### Step 3: Analyze the relationship between the slopes
- **Parallel lines** have equal slopes.
- **Perpendicular lines** have slopes that are negative reciprocals of each other (i.e., \( m_1 \times m_2 = -1 \)).

The slope of \( L_1 \) is \( 2 \), and the slope of \( L_2 \) is \( -\frac{1}{2} \).

Since \( 2 \times \left(-\frac{1}{2}\right) = -1 \), the lines are **perpendicular**.

### Conclusion:
The lines \( L_1 \) and \( L_2 \) are **perpendicular**.

Would you like further details or clarifications? Here are five related questions to explore this concept further:

1. How do we verify if two lines are parallel using their slopes?
2. What happens if both lines have undefined slopes (vertical lines)?
3. Can two lines be neither parallel nor perpendicular? How would you determine that?
4. What is the geometric interpretation of perpendicular lines in a coordinate plane?
5. How do we find the equation of a line using a point and a slope?

**Tip:** Perpendicular slopes are always negative reciprocals. If one slope is \( m \), the other must be \( -\frac{1}{m} \).

Learn how to determine whether two lines are parallel, perpendicular, or neither by calculating their slopes. This detailed solution involves points from lines L1: (2,5), (4,9) and L2: (-1,4), (3,2). Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation