Introduction

When given the slopes of the sides of a triangle, it is possible to determine the angles between the sides. This process involves finding the angles using the arctangent function and understanding the relationships between the slopes and the angles. This article will guide you through the step-by-step process using the given slopes {3, 1/2, -2}.

Understanding the Problem

Given the slopes of the sides of a triangle as 3, 1/2, -2, we aim to find the angles at each vertex of the triangle. The slopes give us the tangents of the angles made by each side with the horizontal axis.

Step 1: Finding Individual Angles

The first step is to find the individual angles corresponding to each slope. These angles can be calculated using the arctangent function.

Step 1.1: Calculate the first angle

For m1 3

θ1 tan-13 ≈ 71.57°

Step 1.2: Calculate the second angle

For m2 1/2

θ2 tan-1(1/2) ≈ 26.57°

Step 1.3: Calculate the third angle

For m3 -2

θ3 tan-1(-2) ≈ -63.43°

Since θ3 is negative, we convert it to a positive angle by adding 180°:

θ3 180° - 63.43° ≈ 116.57°

Step 2: Sum of Angles in a Triangle

Given that the sum of the interior angles in a triangle should equal 180°, we need to verify that the sum of our angles is correct. If not, we need to adjust the angles.

Sum of angles: 71.57° 26.57° 116.57° 214.71°

Clearly, the angles need to be adjusted. Therefore, we calculate the angles between the sides using the arctangent of the differences in slopes.

Step 3: Finding Angles Between Sides

The formula for the angle between two lines with slopes m1 and m2 is:

tanθ (m 1 - m2) / (1 m1 m2)

Step 3.1: Calculate the angle between side 1 and side 2

tanθ12 (3 - 0.5) / (1 3 * 0.5) (2.5) / 2.5 1

θ12 tan-11 45°

Step 3.2: Calculate the angle between side 2 and side 3

tanθ23 (0.5 - (-2)) / (1 0.5 * (-2)) (0.5 2) / (1 - 1) undefined (vertical line)

This indicates that the angle is 90°.

Step 3.3: Calculate the angle between side 1 and side 3

tanθ13 (3 - (-2)) / (1 3 * (-2)) (5) / (-5) -1

θ13 tan-1(-1) 45°

Since tanθ is negative, the angle is 180° - 45° 135° (but this calculation is not necessary since we already have the vertical line case).

Final Angles of the Triangle

Thus, the angles of the triangle are:

Conclusion

In conclusion, the angles of the triangle are approximately 45°, 90°, and 45°. This method of using slopes and the arctangent function can be applied to similar problems in geometry, providing a valuable tool for understanding the relationships between slopes and angles in triangular geometry.

Keywords: triangle angles, slope, arctangent