Introduction

Tackling geometric problems, such as calculating the angle between two lines, is a fundamental skill in mathematics. This article delves into the methods used to determine the angle between two given lines, using a specific example as a guide. We will explore how to deduce the slope of each line, calculate the tangent of the angle between them, and find the acute and obtuse angles. By the end of this article, you will have a thorough understanding of how to solve similar problems.

Methods and Formulas

When calculating the angle between two lines, the primary tools are the slopes of the lines concerned. The slope, denoted as (m), can be calculated using the formula:

[text{Slope} m frac{y_2 - y_1}{x_2 - x_1}]

This formula can be applied to both lines to find their respective slopes. The angle (theta) between two lines with slopes (m_1) and (m_2) can be found using the following tangent formula:

[tan theta left| frac{m_1 - m_2}{1 m_1 m_2} right|]

Example: Calculating the Angle Between Given Lines

In this example, we are given two sets of coordinates. The first line passes through the points (15, 23) and (2, 1), while the second line passes through the points (-2, 3) and (-12, -2).

Let’s begin by calculating the slopes for both lines:

First Line

The coordinates for the first line are (15, 23) and (2, 1).

[text{Slope of the first line} m_1 frac{23 - 1}{15 - 2} frac{22}{13} approx 1.692]

Second Line

The coordinates for the second line are (-2, 3) and (-12, -2).

[text{Slope of the second line} m_2 frac{-2 - 3}{-12 - (-2)} frac{-5}{-10} frac{1}{2}]

Now, we need to use the tangent formula to find the angle between them:

[tan theta left| frac{m_1 - m_2}{1 m_1 m_2} right|]

Substituting the values of (m_1) and (m_2):

[tan theta left| frac{1.692 - 0.5}{1 1.692 cdot 0.5} right| approx left| frac{1.192}{1.846} right| approx 0.645]

Using the inverse tangent function, we find:

[theta approx tan^{-1} (0.645) approx 33.05^circ]

Since (theta) is the angle between the lines, and the angle between two lines can be either acute or obtuse, we should also consider the obtuse angle, which is:

[text{180}^circ - theta approx 180^circ - 33.05^circ approx 146.95^circ]

Conclusion

Understanding how to calculate the angle between two lines is essential for solving various geometric and analytical problems. By applying the slope formula and the tangent of the angle between the lines, we can determine both the acute and obtuse angles. This method not only provides a step-by-step approach but also helps in visualizing the geometric relationship between the lines.

For further practice and more detailed explanations on this and other mathematical concepts, please refer to the resources and additional learning materials at the end of this article.