Finding the Length of the Line Segment

Oct 13, 2025 by windailys.com 3 views

Introduction

In mathematics, a directed line segment is a line segment with a specific direction. It is defined by two points, known as the initial point and the terminal point. In this article, we will explore the concept of a directed line segment and how to find the location of a point that is a fraction of the length of the line segment.

The Directed Line Segment

The directed line segment in this problem goes from point B(6, -2) to point C(-4, 3). This means that the initial point is B(6, -2) and the terminal point is C(-4, 3). The length of the line segment can be found using the distance formula:

Distance Formula

The distance formula is used to find the distance between two points in a coordinate plane. It is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Finding the Length of the Line Segment

To find the length of the line segment, we can use the distance formula:

d = √((-4 - 6)^2 + (3 - (-2))^2) d = √((-10)^2 + (5)^2) d = √(100 + 25) d = √125

Finding the Location of Point P

Point P is 4/5 the length of the line from B to C. This means that the distance from B to P is 4/5 of the length of the line segment, and the distance from P to C is 1/5 of the length of the line segment.

Finding the Coordinates of Point P

To find the coordinates of point P, we need to find the point that is 4/5 of the length of the line segment from B to C. We can do this by using the section formula.

Section Formula

The section formula is used to find the coordinates of a point that divides a line segment into a particular ratio. It is given by:

(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

where (x1, y1) and (x2, y2) are the coordinates of the two points, and m/n is the ratio.

Applying the Section Formula

In this case, we want to find the point that is 4/5 of the length of the line segment from B to C. This means that the ratio is 4/5, and the coordinates of the two points are B(6, -2) and C(-4, 3).

(x, y) = ((4/5)(-4) + (1/5)(6), (4/5)(3) + (1/5)(-2)) (x, y) = ((-16/5) + (6/5), (12/5) + (-2/5)) (x, y) = (-10/5, 10/5) (x, y) = (-2, 2)

Conclusion

In this article, we explored the concept of a directed line segment and how to find the location of a point that is a fraction of the length of the line segment. We used the distance formula to find the length of the line segment, and the section formula to find the coordinates of point P. The coordinates of point P are (-2, 2).

Final Answer

The final answer is (-2, 2).

Introduction

In our previous article, we explored the concept of a directed line segment and how to find the location of a point that is a fraction of the length of the line segment. In this article, we will answer some of the most frequently asked questions related to this topic.

Q&A

Q: What is a directed line segment?

A: A directed line segment is a line segment with a specific direction. It is defined by two points, known as the initial point and the terminal point.

Q: How do I find the length of a directed line segment?

A: To find the length of a directed line segment, you can use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: What is the section formula?

A: The section formula is used to find the coordinates of a point that divides a line segment into a particular ratio. It is given by:

(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

where (x1, y1) and (x2, y2) are the coordinates of the two points, and m/n is the ratio.

Q: How do I find the location of a point that is a fraction of the length of a line segment?

A: To find the location of a point that is a fraction of the length of a line segment, you can use the section formula. You need to find the point that divides the line segment into the desired ratio.

Q: What is the difference between the distance formula and the section formula?

A: The distance formula is used to find the length of a line segment, while the section formula is used to find the coordinates of a point that divides a line segment into a particular ratio.

Q: Can I use the section formula to find the length of a line segment?

A: No, the section formula is not used to find the length of a line segment. It is used to find the coordinates of a point that divides a line segment into a particular ratio.

Q: How do I apply the section formula in real-life situations?

A: The section formula has many real-life applications, such as finding the coordinates of a point that divides a line segment into a particular ratio in geometry, trigonometry, and engineering.

Q: Can I use the section formula to find the coordinates of a point that is not a fraction of the length of a line segment?

A: No, the section formula is used to find the coordinates of a point that divides a line segment into a particular ratio. If you want to find the coordinates of a point that is not a fraction of the length of a line segment, you need to use a different formula or method.

Conclusion

In this article, we answered some of the most frequently asked questions related to the concept of a directed line segment and how to find the location of a point that is a fraction of the length of the line segment. We hope that this article has provided you with a better understanding of this topic and has helped you to apply the section formula in real-life situations.

Final Answer

The final answer is that the section formula is a powerful tool that can be used to find the coordinates of a point that divides a line segment into a particular ratio. It has many real-life applications and can be used to solve a wide range of problems in geometry, trigonometry, and engineering.