Find the slope and inclination of the line passing through the two given points.

a) A(3,-7), B(4, 8); b) A(9,0), B(3, 18)

### a) Points: \( A(3, -7) \) and \( B(4, 8) \)

**1. Slope (m):**

The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

For \( A(3, -7) \) and \( B(4, 8) \):

\[
m = \frac{8 - (-7)}{4 - 3} = \frac{8 + 7}{1} = \frac{15}{1} = 15
\]

**2. Inclination (θ):**

The inclination of a line is the angle \( θ \) it makes with the positive direction of the x-axis. It can be found using:

\[
\theta = \tan^{-1}(m)
\]

\[
\theta = \tan^{-1}(15)
\]

Using a calculator, the inclination is approximately:

\[
\theta \approx 86.2^\circ
\]

---

### b) Points: \( A(9, 0) \) and \( B(3, 18) \)

**1. Slope (m):**

Using the same formula for slope:

\[
m = \frac{18 - 0}{3 - 9} = \frac{18}{-6} = -3
\]

**2. Inclination (θ):**

Now, find the inclination:

\[
\theta = \tan^{-1}(-3)
\]

Using a calculator:

\[
\theta \approx -71.57^\circ
\]

However, inclination is usually given as a positive angle, so:

\[
\theta = 180^\circ + (-71.57^\circ) = 108.43^\circ
\]

---

### Final Answer:

- a) Slope = 15, Inclination \( \theta \approx 86.2^\circ \)
- b) Slope = -3, Inclination \( \theta \approx 108.43^\circ \)

Do you need more details or have any questions?

---

**Relative Questions:**

1. How is the slope affected by vertical or horizontal lines?
2. What is the slope of a line that passes through two points with the same x-coordinate?
3. How do you calculate the slope and inclination of parallel lines?
4. How does the inclination change if the slope is negative?
5. What would the inclination be for a line with a slope of 0?

**Tip:** The inclination is always measured counterclockwise from the positive x-axis.

Find the slope and inclination of the line passing through the two given points. a) A(3,-7), B(4, 8); b) A(9,0), B(3, 18)

Learn how to calculate the slope and inclination of a line passing through two given points with a detailed explanation. This problem includes step-by-step guidance on using the slope formula and calculating the inclination using inverse trigonometric functions. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation