https://r2.erweima.ai/mathbot/image/1724456396020f9eonezm.jpg

Let's solve the problem step-by-step based on the given image.

### Problem Summary

We are given three lines that pass through specific points:

- **Line 1** passes through \((2, 7)\) and \((-3, 8)\).
- **Line 2** passes through \((3, -3)\) and \((1, 5)\).
- **Line 3** passes through \((-3, 0)\) and \((1, 8)\).

**(a) Find the slope of each line.**

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Let's calculate the slope for each line.

#### Slope of Line 1:
Points: \((2, 7)\) and \((-3, 8)\)

\[
m_1 = \frac{8 - 7}{-3 - 2} = \frac{1}{-5} = -\frac{1}{5}
\]

#### Slope of Line 2:
Points: \((3, -3)\) and \((1, 5)\)

\[
m_2 = \frac{5 - (-3)}{1 - 3} = \frac{5 + 3}{1 - 3} = \frac{8}{-2} = -4
\]

#### Slope of Line 3:
Points: \((-3, 0)\) and \((1, 8)\)

\[
m_3 = \frac{8 - 0}{1 - (-3)} = \frac{8}{1 + 3} = \frac{8}{4} = 2
\]

**Slopes:**
- Slope of Line 1: \(-\frac{1}{5}\)
- Slope of Line 2: \(-4\)
- Slope of Line 3: \(2\)

**(b) Determine if the pair of lines are parallel, perpendicular, or neither.**

- **Lines are parallel** if their slopes are equal.
- **Lines are perpendicular** if the product of their slopes is \(-1\).

#### Checking each pair:

1. **Line 1 and Line 2:**
   - Slopes: \(-\frac{1}{5}\) and \(-4\)
   - \(-\frac{1}{5} \times (-4) = \frac{4}{5}\), which is not \(-1\).
   - **Conclusion:** Neither parallel nor perpendicular.

2. **Line 1 and Line 3:**
   - Slopes: \(-\frac{1}{5}\) and \(2\)
   - \(-\frac{1}{5} \times 2 = -\frac{2}{5}\), which is not \(-1\).
   - **Conclusion:** Neither parallel nor perpendicular.

3. **Line 2 and Line 3:**
   - Slopes: \(-4\) and \(2\)
   - \(-4 \times 2 = -8\), which is not \(-1\).
   - **Conclusion:** Neither parallel nor perpendicular.

### Final Answers:

1. **Slopes:**
   - Line 1: \(-\frac{1}{5}\)
   - Line 2: \(-4\)
   - Line 3: \(2\)

2. **Pairwise Relationship:**
   - Line 1 and Line 2: Neither parallel nor perpendicular.
   - Line 1 and Line 3: Neither parallel nor perpendicular.
   - Line 2 and Line 3: Neither parallel nor perpendicular.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you determine the equation of a line given two points?
2. What is the geometric interpretation of a line's slope?
3. How can you find the distance between two points on a coordinate plane?
4. What are the conditions for three points to be collinear?
5. How can you determine if a point lies on a given line?

**Tip:** Always double-check your calculations when determining the slope, as small arithmetic mistakes can lead to incorrect conclusions about the lines' relationships.

Explore how to analyze the relationships between lines based on their slopes. Step-by-step calculations and conclusions are provided for determining whether lines are parallel, perpendicular, or neither. Suitable for 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