What Are The { X $}$- And { Y $}$-coordinates Of Point { P $}$ On The Directed Line Segment From { K $}$ To { J $}$ Such That { P $}$ Is { \frac{3}{5}$}$ The Length Of The Line

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Introduction

In geometry, a line segment is a part of a line that is bounded by two distinct end points. A directed line segment is a line segment with a specific direction, which is often represented by an arrow. In this article, we will discuss how to find the coordinates of a point on a directed line segment such that the point is a certain fraction of the length of the line segment.

Understanding the Problem

Let's consider a directed line segment from point { K $}$ to point { J $}$. We want to find the coordinates of a point { P $}$ on this line segment such that { P $}$ is {\frac{3}{5}$}$ the length of the line segment. This means that the distance from { K $}$ to { P $}$ is {\frac{3}{5}$}$ of the distance from { K $}$ to { J $}$.

Using Similar Triangles

To solve this problem, we can use the concept of similar triangles. Let's draw a diagram of the line segment and the point { P $}$.

  +---------------+
  |              |
  |  K  ------>  |
  |              |
  +---------------+
           |
           |
           v
  +---------------+
  |              |
  |  P  ------>  |
  |              |
  +---------------+
           |
           |
           v
  +---------------+
  |              |
  |  J  ------>  |
  |              |
  +---------------+

In this diagram, we can see that the triangle formed by points { K $}$, { P $}$, and { J $}$ is similar to the triangle formed by points { K $}$, { J $}$, and the origin. This is because the two triangles have the same angles.

Finding the Coordinates of Point { P $}$

Since the two triangles are similar, we can set up a proportion to find the coordinates of point { P $}$. Let's say the coordinates of point { K $}$ are { (x_1, y_1) $}$ and the coordinates of point { J $}$ are { (x_2, y_2) $}$. Then, the coordinates of point { P $}$ are { (x, y) $}$.

We can set up the following proportion:

x−x1x2−x1=35\frac{x - x_1}{x_2 - x_1} = \frac{3}{5}

y−y1y2−y1=35\frac{y - y_1}{y_2 - y_1} = \frac{3}{5}

Solving for { x $}$ and { y $}$, we get:

x=x1+35(x2−x1)x = x_1 + \frac{3}{5}(x_2 - x_1)

y=y1+35(y2−y1)y = y_1 + \frac{3}{5}(y_2 - y_1)

Example

Let's consider an example to illustrate how to use this formula. Suppose the coordinates of point { K $}$ are { (2, 3) $}$ and the coordinates of point { J $}$ are { (6, 9) $}$. We want to find the coordinates of point { P $}$ such that { P $}$ is {\frac{3}{5}$}$ the length of the line segment.

Using the formula, we get:

x=2+35(6−2)x = 2 + \frac{3}{5}(6 - 2)

x=2+35(4)x = 2 + \frac{3}{5}(4)

x=2+125x = 2 + \frac{12}{5}

x=105+125x = \frac{10}{5} + \frac{12}{5}

x=225x = \frac{22}{5}

y=3+35(9−3)y = 3 + \frac{3}{5}(9 - 3)

y=3+35(6)y = 3 + \frac{3}{5}(6)

y=3+185y = 3 + \frac{18}{5}

y=155+185y = \frac{15}{5} + \frac{18}{5}

y=335y = \frac{33}{5}

Therefore, the coordinates of point { P $}$ are { (\frac{22}{5}, \frac{33}{5}) $}$.

Conclusion

In this article, we discussed how to find the coordinates of a point on a directed line segment such that the point is a certain fraction of the length of the line segment. We used the concept of similar triangles and set up a proportion to find the coordinates of the point. We also provided an example to illustrate how to use this formula. This formula can be used in a variety of applications, such as finding the coordinates of a point on a line segment that is a certain fraction of the length of the line segment.

References

Further Reading

Introduction

In our previous article, we discussed how to find the coordinates of a point on a directed line segment such that the point is a certain fraction of the length of the line segment. In this article, we will answer some frequently asked questions about this topic.

Q: What is the formula for finding the coordinates of a point on a directed line segment?

A: The formula for finding the coordinates of a point on a directed line segment is:

x=x1+mn(x2−x1)x = x_1 + \frac{m}{n}(x_2 - x_1)

y=y1+mn(y2−y1)y = y_1 + \frac{m}{n}(y_2 - y_1)

where { (x_1, y_1) $}$ are the coordinates of the starting point, { (x_2, y_2) $}$ are the coordinates of the ending point, and { m $}$ and { n $}$ are the numerator and denominator of the fraction, respectively.

Q: What is the significance of the fraction {\frac{m}{n}$) in the formula?

A: The fraction [\frac{m}{n}\$\) represents the fraction of the length of the line segment that the point is from the starting point. For example, if \[\frac{m}{n}$) is [\frac{3}{5}\$\) , then the point is \[\frac{3}{5}$) of the length of the line segment from the starting point.

Q: How do I determine the coordinates of the starting and ending points?

A: The coordinates of the starting and ending points are given in the problem statement. For example, if the problem states that the starting point is [$ (2, 3) $}$ and the ending point is { (6, 9) $}$, then you can use these coordinates in the formula.

Q: What if the fraction [\frac{m}{n}\$\) is not a simple fraction, such as \[\frac{3}{7}$)?

A: If the fraction [$\frac{m}{n}$) is not a simple fraction, you can still use the formula. However, you will need to simplify the fraction before plugging it into the formula.

Q: Can I use this formula to find the coordinates of a point on a line segment that is not directed?

A: Yes, you can use this formula to find the coordinates of a point on a line segment that is not directed. However, you will need to use the formula for the midpoint of the line segment instead of the formula for the point on the line segment.

Q: What if I make a mistake in the formula or in the calculations?

A: If you make a mistake in the formula or in the calculations, you may get incorrect coordinates for the point. To avoid this, make sure to double-check your work and use a calculator or computer program to check your answers.

Q: Can I use this formula to find the coordinates of a point on a line segment that is not a straight line?

A: No, you cannot use this formula to find the coordinates of a point on a line segment that is not a straight line. This formula is only valid for straight line segments.

Q: What if I need to find the coordinates of a point on a line segment that is not in the first quadrant?

A: If you need to find the coordinates of a point on a line segment that is not in the first quadrant, you will need to use a different formula or method. This formula is only valid for line segments in the first quadrant.

Conclusion

In this article, we answered some frequently asked questions about finding the coordinates of a point on a directed line segment. We hope that this article has been helpful in clarifying any confusion you may have had about this topic.

References

Further Reading