What Are The $x$- And $y$-coordinates Of Point $E$, Which Partitions The Directed Line Segment From $I$ To $K$ Into A Ratio Of $1:4$?$[ x = \left(\frac{m}{m+n}\right)(x_2-x_1) +

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Introduction

When dealing with directed line segments and ratios, it's essential to understand how to find the coordinates of a point that partitions the line segment into a specific ratio. In this case, we're given a directed line segment from point $I$ to point $K$, and we need to find the coordinates of point $E$, which partitions the line segment into a ratio of $1:4$. To solve this problem, we'll use the concept of section formula, which allows us to find the coordinates of a point that divides a line segment into a given ratio.

Section Formula

The section formula states that if a point $P(x, y)$ divides the line segment joining the points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, then the coordinates of point $P$ are given by:

x=(mm+n)(x2−x1)+x1x = \left(\frac{m}{m+n}\right)(x_2-x_1) + x_1

y=(mm+n)(y2−y1)+y1y = \left(\frac{m}{m+n}\right)(y_2-y_1) + y_1

Applying the Section Formula

In this case, we're given that point $E$ partitions the directed line segment from point $I$ to point $K$ in the ratio $1:4$. We can use the section formula to find the coordinates of point $E$.

Let's assume the coordinates of point $I$ are $(x_1, y_1)$ and the coordinates of point $K$ are $(x_2, y_2)$. Since point $E$ partitions the line segment in the ratio $1:4$, we can use the section formula to find the coordinates of point $E$.

Calculating the $x$-coordinate of point $E$

Using the section formula, we can calculate the $x$-coordinate of point $E$ as follows:

x=(11+4)(x2−x1)+x1x = \left(\frac{1}{1+4}\right)(x_2-x_1) + x_1

x=(15)(x2−x1)+x1x = \left(\frac{1}{5}\right)(x_2-x_1) + x_1

Calculating the $y$-coordinate of point $E$

Similarly, we can calculate the $y$-coordinate of point $E$ as follows:

y=(11+4)(y2−y1)+y1y = \left(\frac{1}{1+4}\right)(y_2-y_1) + y_1

y=(15)(y2−y1)+y1y = \left(\frac{1}{5}\right)(y_2-y_1) + y_1

Example

Let's consider an example to illustrate how to use the section formula to find the coordinates of point $E$.

Suppose the coordinates of point $I$ are $(2, 3)$ and the coordinates of point $K$ are $(6, 9)$. We need to find the coordinates of point $E$, which partitions the directed line segment from point $I$ to point $K$ in the ratio $1:4$.

Using the section formula, we can calculate the $x$-coordinate of point $E$ as follows:

x=(11+4)(6−2)+2x = \left(\frac{1}{1+4}\right)(6-2) + 2

x=(15)(4)+2x = \left(\frac{1}{5}\right)(4) + 2

x=45+2x = \frac{4}{5} + 2

x=45+105x = \frac{4}{5} + \frac{10}{5}

x=145x = \frac{14}{5}

Similarly, we can calculate the $y$-coordinate of point $E$ as follows:

y=(11+4)(9−3)+3y = \left(\frac{1}{1+4}\right)(9-3) + 3

y=(15)(6)+3y = \left(\frac{1}{5}\right)(6) + 3

y=65+3y = \frac{6}{5} + 3

y=65+155y = \frac{6}{5} + \frac{15}{5}

y=215y = \frac{21}{5}

Therefore, the coordinates of point $E$ are $(\frac{14}{5}, \frac{21}{5})$.

Conclusion

In this article, we discussed how to find the coordinates of a point that partitions a directed line segment into a specific ratio using the section formula. We applied the section formula to find the coordinates of point $E$, which partitions the directed line segment from point $I$ to point $K$ in the ratio $1:4$. We also provided an example to illustrate how to use the section formula in practice. By understanding the section formula and how to apply it, we can solve a wide range of problems involving directed line segments and ratios.

Frequently Asked Questions

  • What is the section formula? The section formula is a mathematical formula that allows us to find the coordinates of a point that divides a line segment into a given ratio.
  • How do I apply the section formula to find the coordinates of a point? To apply the section formula, you need to know the coordinates of the two points that define the line segment, as well as the ratio in which the point divides the line segment.
  • What is the ratio in which point $E$ divides the line segment from point $I$ to point $K$? The ratio in which point $E$ divides the line segment from point $I$ to point $K$ is $1:4$.

Further Reading

  • Section Formula: A Comprehensive Guide
  • Directed Line Segments: A Mathematical Perspective
  • Ratios and Proportions: A Mathematical Introduction

Introduction

In our previous article, we discussed how to find the coordinates of a point that partitions a directed line segment into a specific ratio using the section formula. In this article, we'll answer some frequently asked questions about the section formula and directed line segments.

Q&A

Q: What is the section formula?

A: The section formula is a mathematical formula that allows us to find the coordinates of a point that divides a line segment into a given ratio.

Q: How do I apply the section formula to find the coordinates of a point?

A: To apply the section formula, you need to know the coordinates of the two points that define the line segment, as well as the ratio in which the point divides the line segment. The section formula is given by:

x=(mm+n)(x2−x1)+x1x = \left(\frac{m}{m+n}\right)(x_2-x_1) + x_1

y=(mm+n)(y2−y1)+y1y = \left(\frac{m}{m+n}\right)(y_2-y_1) + y_1

Q: What is the ratio in which point $E$ divides the line segment from point $I$ to point $K$?

A: The ratio in which point $E$ divides the line segment from point $I$ to point $K$ is $1:4$.

Q: Can I use the section formula to find the coordinates of a point that divides a line segment into a ratio of $2:3$?

A: Yes, you can use the section formula to find the coordinates of a point that divides a line segment into a ratio of $2:3$. Simply substitute $m=2$ and $n=3$ into the section formula.

Q: What if I don't know the coordinates of the two points that define the line segment?

A: If you don't know the coordinates of the two points that define the line segment, you won't be able to use the section formula to find the coordinates of a point that divides the line segment into a given ratio.

Q: Can I use the section formula to find the coordinates of a point that divides a line segment into a ratio of $3:2$?

A: Yes, you can use the section formula to find the coordinates of a point that divides a line segment into a ratio of $3:2$. Simply substitute $m=3$ and $n=2$ into the section formula.

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

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

Q: Can I use the section formula to find the coordinates of a point that divides a line segment into a ratio of $1:1$?

A: Yes, you can use the section formula to find the coordinates of a point that divides a line segment into a ratio of $1:1$. In this case, the section formula simplifies to:

x=x1+x22x = \frac{x_1 + x_2}{2}

y=y1+y22y = \frac{y_1 + y_2}{2}

Conclusion

In this article, we answered some frequently asked questions about the section formula and directed line segments. We hope this article has been helpful in clarifying any confusion you may have had about the section formula and its applications.

Further Reading

  • Section Formula: A Comprehensive Guide
  • Directed Line Segments: A Mathematical Perspective
  • Ratios and Proportions: A Mathematical Introduction

Additional Resources

  • Khan Academy: Section Formula
  • Mathway: Section Formula
  • Wolfram Alpha: Section Formula

Related Articles

  • Finding the Coordinates of a Point that Divides a Line Segment into a Given Ratio
  • The Midpoint Formula: A Mathematical Introduction
  • Ratios and Proportions: A Mathematical Perspective