When dealing with angles and slopes in coordinate geometry, it's essential to understand how the angle affects the slope of a line. In this article, we will go through the steps to find the slope of a line that makes an angle of 135 degrees with the negative direction of the X-axis. We'll provide a detailed explanation with visual aids, ensuring a comprehensive understanding of the topic.

Understanding the Angle

The angle of 135 degrees is measured counterclockwise from the negative X-axis. This means we need to visualize this angle correctly. The negative X-axis points to the left, and we move counterclockwise to obtain the angle of 135 degrees.

Step 1: Understand the Angle

Let's break down the angle step-by-step:

The negative X-axis points to the left, i.e., 180 degrees from the positive X-axis. Starting from the negative X-axis, moving counterclockwise to 135 degrees means we need to move an additional 45 degrees to reach 135 degrees.

Therefore, when measured in standard position from the positive X-axis, the angle is:

180° - 135° 45°

Step 2: Convert the Angle to Standard Position

Angles in standard position are measured counterclockwise from the positive X-axis. The equivalent angle in standard position is 45 degrees.

Step 3: Calculate the Slope

The slope m of a line can be calculated using the tangent of the angle it forms with the positive X-axis:

m tan(θ)

Here, θ 45°. Therefore:

m tan(45°) 1

Conclusion

The slope of the line that makes an angle of 135 degrees with the negative direction of the X-axis is 1. This means the line has a 45-degree angle with the positive X-axis and rises one unit vertically for every one unit it moves horizontally to the right.

Visualization

For a better understanding, let's visualize this on a coordinate plane:

Draw the X-axis and Y-axis. Mark the negative X-axis. From the negative X-axis, measure 135 degrees counterclockwise. The resulting line will have a slope of 1.

Summary

Angle with the negative X-axis: 135 degrees

Equivalent angle with the positive X-axis: 45 degrees

Slope of the line: tan(45°) 1

If you have any further questions or need additional explanations, feel free to ask!

Note: Depending on the direction of measurement, the slope can either be 1 or -1. The choice depends on the reference direction.

Alternatively, we can simplify 135 degrees as:

135° 90° 45°

This indicates that the ray makes a 45-degree angle with the positive X-axis.