Understanding the Angle Between Two Lines Through a Point Using Slopes

When dealing with geometric problems involving the angle between two lines passing through a common point, a fundamental concept in geometry and linear algebra, the slopes of the lines play a crucial role. In this article, we will explore how to calculate the angle between two lines that pass through a point, specifically focusing on the origin (0,0), using the slope formula.

Problem Statement and Approach

The specific problem at hand involves calculating the angle between the line segment joining points (5, 2) and (6, -15) with respect to the origin (0, 0).

Step 1: Determine the Slopes of the Lines

First, we need to find the slopes of the lines formed by connecting the origin to the given points. The slope of a line is given by the formula:

m (y? - y?) / (x? - x?)

Slope of Line from Origin to Point A (5, 2)

The slope (m?) of the line from the origin (0, 0) to the point (5, 2) is:

m? (2 - 0) / (5 - 0) 2 / 5 0.4

Slope of Line from Origin to Point B (6, -15)

The slope (m?) of the line from the origin (0, 0) to the point (6, -15) is:

m? (-15 - 0) / (6 - 0) -15 / 6 -2.5

Step 2: Use the Slopes to Calculate the Angle Between the Lines

To find the angle (θ) between two lines with slopes m? and m?, we use the formula:

tanθ (m? - m?) / (1 m?m?)

Applying the Formulas

Substituting the values of m? and m? into the formula:

tanθ (0.4 - (-2.5)) / (1 (0.4) * (-2.5))

Simplifying the expression:

tanθ (0.4 2.5) / (1 - 1) 2.9 / 0

The result is undefined, indicating that the lines are perpendicular. Therefore, the angle between them is:

θ 90°

Conclusion

The angle made by the line joining the points (5, 2) and (6, -15) with respect to the origin (0, 0) is 90 degrees. This is because the lines are perpendicular, and the tangent function results in an undefined value, confirming this relationship.

Additional Insights

Understanding the concept of slopes and their relationship to angles between lines is important in various fields such as calculus, physics, and computer graphics. The method discussed here can be applied to any pair of lines that intersect at a point, not just the origin, by adjusting the coordinates accordingly.