How to Calculate the Line Segment Joining Two Specific Points

Introduction

In geometry, the line segment is a fundamental concept. Determining the line segment joining two specific points can be achieved through several steps, which include identifying the points, calculating the midpoint, finding the slope, and finally deriving the equation of the line segment. This article provides a comprehensive guide on how to find the line segment joining the points (a 1, 2a 3) and (a - 1, 2a - 1).

Identifying the Points

Let the points be:

(P_1 (a 1, 2a 3)) (P_2 (a - 1, 2a - 1))

Calculating the Midpoint

The midpoint (M) of the line segment can be found using the midpoint formula:

(M left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right))

Substituting the coordinates of the points:

(M left(frac{a 1 a - 1}{2}, frac{2a 3 2a - 1}{2}right))

(M left(frac{2a}{2}, frac{4a 2}{2}right)) (left(a, 2a 1right))

Finding the Slope

The slope (m) of the line segment can be calculated using the slope formula:

(m frac{y_2 - y_1}{x_2 - x_1})

Substituting the coordinates:

(m frac{(2a - 1) - (2a 3)}{(a - 1) - (a 1)} frac{-4}{-2} 2)

Deriving the Equation of the Line Segment

Using the point-slope form of the equation of a line (y - y_1 m(x - x_1)), we can use point (P_1 (a, 2a 3)) to write the equation:

(y - (2a 3) 2(x - a))

(y - 2a 3 2x - 2a)

(y 2x 1)

Summary

Midpoint: (M (a, 2a 1)) Slope: (m 2) Equation of Line Segment: (y 2x 1)

This gives you the line segment joining the two points in terms of (a).

Length of the Line Segment

While the above steps give us the equation of the line, you may also want to know the length of the line segment joining these points. The distance formula in a 2D plane is given by:

(d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2})

Calculating the distance:

(delta x |a 1 - (a - 1)| 2)

(delta y |2a 3 - (2a - 1)| 4)

(d sqrt{(delta x)^2 (delta y)^2} sqrt{2^2 4^2} sqrt{4 16} sqrt{20} 2sqrt{5})

The length of the line segment joining points (P_1 (a 1, 2a 3)) and (P_2 (a - 1, 2a - 1)) is (2sqrt{5}).

Conclusion

The process of finding the line segment joining two specific points involves calculating the midpoint, finding the slope, and deriving the equation of the line segment. Additionally, the length of the segment can be calculated using the distance formula. Understanding these steps is crucial for analyzing and working with line segments in geometry.