What Is The { X$}$-coordinate Of The Point That Divides The Directed Line Segment From { J$}$ To { K$}$ Into A Ratio Of 2:5?${ X = \left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 }$A. { -4$}$B.

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Introduction to Section Formula

The section formula is a fundamental concept in mathematics that helps us find the coordinates of a point that divides a line segment into a particular ratio. In this article, we will explore the section formula and use it to find the {x$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of 2:5.

Understanding the Section Formula

The section formula is given by the equation:

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

where {x$}$ is the {x$}$-coordinate of the point that divides the line segment, {m$}$ and {n$}$ are the ratios in which the line segment is divided, and {x_1$}$ and {x_2$}$ are the {x$}$-coordinates of the endpoints of the line segment.

Applying the Section Formula

To find the {x$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of 2:5, we can use the section formula. Let's assume that the {x$}$-coordinates of points {J$}$ and {K$}$ are {x_1$}$ and {x_2$}$, respectively. We want to find the {x$}$-coordinate of the point that divides the line segment in a ratio of 2:5, so we can set up the following equation:

x=(22+5)(x2−x1)+x1{ x = \left(\frac{2}{2+5}\right)\left(x_2-x_1\right)+x_1 }

Simplifying the equation, we get:

x=(27)(x2−x1)+x1{ x = \left(\frac{2}{7}\right)\left(x_2-x_1\right)+x_1 }

Finding the {x$}$-coordinate

To find the {x$}$-coordinate of the point that divides the line segment, we need to know the values of {x_1$}$ and {x_2$}$. However, the problem does not provide these values. Therefore, we cannot find the exact value of the {x$}$-coordinate.

Conclusion

In this article, we explored the section formula and used it to find the {x$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of 2:5. We set up the equation using the section formula and simplified it to find the {x$}$-coordinate. However, we were unable to find the exact value of the {x$}$-coordinate due to the lack of information about the endpoints of the line segment.

Example Problems

Problem 1

Find the {x$}$-coordinate of the point that divides the directed line segment from {A$}$ to {B$}$ into a ratio of 3:4.

Solution

To find the {x$}$-coordinate of the point that divides the line segment, we can use the section formula. Let's assume that the {x$}$-coordinates of points {A$}$ and {B$}$ are {x_1$}$ and {x_2$}$, respectively. We want to find the {x$}$-coordinate of the point that divides the line segment in a ratio of 3:4, so we can set up the following equation:

x=(33+4)(x2−x1)+x1{ x = \left(\frac{3}{3+4}\right)\left(x_2-x_1\right)+x_1 }

Simplifying the equation, we get:

x=(37)(x2−x1)+x1{ x = \left(\frac{3}{7}\right)\left(x_2-x_1\right)+x_1 }

Problem 2

Find the {x$}$-coordinate of the point that divides the directed line segment from {C$}$ to {D$}$ into a ratio of 2:3.

Solution

To find the {x$}$-coordinate of the point that divides the line segment, we can use the section formula. Let's assume that the {x$}$-coordinates of points {C$}$ and {D$}$ are {x_1$}$ and {x_2$}$, respectively. We want to find the {x$}$-coordinate of the point that divides the line segment in a ratio of 2:3, so we can set up the following equation:

x=(22+3)(x2−x1)+x1{ x = \left(\frac{2}{2+3}\right)\left(x_2-x_1\right)+x_1 }

Simplifying the equation, we get:

x=(25)(x2−x1)+x1{ x = \left(\frac{2}{5}\right)\left(x_2-x_1\right)+x_1 }

Final Answer

The final answer is not provided as the problem does not provide the necessary information to find the exact value of the {x$}$-coordinate. However, the section formula can be used to find the {x$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of 2:5.

Introduction

In our previous article, we explored the section formula and used it to find the {x$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of 2:5. However, we were unable to find the exact value of the {x$}$-coordinate due to the lack of information about the endpoints of the line segment. In this article, we will answer some frequently asked questions about the section formula and directed line segments.

Q&A

Q1: What is the section formula?

A1: The section formula is a fundamental concept in mathematics that helps us find the coordinates of a point that divides a line segment into a particular ratio. It is given by the equation:

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

where {x$}$ is the {x$}$-coordinate of the point that divides the line segment, {m$}$ and {n$}$ are the ratios in which the line segment is divided, and {x_1$}$ and {x_2$}$ are the {x$}$-coordinates of the endpoints of the line segment.

Q2: How do I use the section formula to find the {x$}$-coordinate of the point that divides a line segment?

A2: To use the section formula, you need to know the values of {m$}$, {n$}$, {x_1$}$, and {x_2$}$. You can then plug these values into the equation:

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

Simplifying the equation, you will get the {x$}$-coordinate of the point that divides the line segment.

Q3: What is the difference between the section formula and the midpoint formula?

A3: The section formula and the midpoint formula are both used to find the coordinates of a point that divides a line segment. However, the section formula is used to find the point that divides the line segment in a particular ratio, while the midpoint formula is used to find the midpoint of the line segment. The midpoint formula is given by the equation:

x=x1+x22{ x = \frac{x_1+x_2}{2} }

Q4: Can I use the section formula to find the {y$}$-coordinate of the point that divides a line segment?

A4: Yes, you can use the section formula to find the {y$}$-coordinate of the point that divides a line segment. The section formula is given by the equation:

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

You can replace {x$}$ with {y$}$ and use the same equation to find the {y$}$-coordinate of the point that divides the line segment.

Q5: What are some real-world applications of the section formula?

A5: The section formula has many real-world applications, including:

  • Finding the coordinates of a point that divides a line segment in a particular ratio
  • Calculating the area of a triangle or a quadrilateral
  • Finding the length of a line segment
  • Solving problems in geometry and trigonometry

Conclusion

In this article, we answered some frequently asked questions about the section formula and directed line segments. We hope that this article has helped you understand the section formula and its applications. If you have any more questions, feel free to ask!

Example Problems

Problem 1

Find the {x$}$-coordinate of the point that divides the directed line segment from {A$}$ to {B$}$ into a ratio of 3:4.

Solution

To find the {x$}$-coordinate of the point that divides the line segment, we can use the section formula. Let's assume that the {x$}$-coordinates of points {A$}$ and {B$}$ are {x_1$}$ and {x_2$}$, respectively. We want to find the {x$}$-coordinate of the point that divides the line segment in a ratio of 3:4, so we can set up the following equation:

x=(33+4)(x2−x1)+x1{ x = \left(\frac{3}{3+4}\right)\left(x_2-x_1\right)+x_1 }

Simplifying the equation, we get:

x=(37)(x2−x1)+x1{ x = \left(\frac{3}{7}\right)\left(x_2-x_1\right)+x_1 }

Problem 2

Find the {y$}$-coordinate of the point that divides the directed line segment from {C$}$ to {D$}$ into a ratio of 2:3.

Solution

To find the {y$}$-coordinate of the point that divides the line segment, we can use the section formula. Let's assume that the {y$}$-coordinates of points {C$}$ and {D$}$ are {y_1$}$ and {y_2$}$, respectively. We want to find the {y$}$-coordinate of the point that divides the line segment in a ratio of 2:3, so we can set up the following equation:

y=(22+3)(y2−y1)+y1{ y = \left(\frac{2}{2+3}\right)\left(y_2-y_1\right)+y_1 }

Simplifying the equation, we get:

y=(25)(y2−y1)+y1{ y = \left(\frac{2}{5}\right)\left(y_2-y_1\right)+y_1 }

Final Answer

The final answer is not provided as the problem does not provide the necessary information to find the exact value of the {x$}$-coordinate or the {y$}$-coordinate. However, the section formula can be used to find the coordinates of a point that divides a line segment in a particular ratio.