What Is The $x$-coordinate Of The Point That Divides The Directed Line Segment From K To J Into A Ratio Of $1:3$?Use The Formula: ${ X = \left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 }$Options:A. -1 B. 3 C. 7 D.

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Introduction

In geometry, a line segment can be divided into a specific ratio by finding the point that divides the segment into that ratio. This concept is crucial in various mathematical applications, including coordinate geometry. In this article, we will explore how to find the $x$-coordinate of the point that divides the directed line segment from K to J into a ratio of $1:3$.

Understanding the Problem

To solve this problem, we need to understand the concept of dividing a line segment into a specific ratio. The ratio of $1:3$ means that the point we are looking for is $1$ part of the way from point K to point J, while the remaining $3$ parts are from the point J to the point K. This can be visualized as a line segment with point K at one end and point J at the other end.

Formula for Dividing a Line Segment

The formula to find the $x$-coordinate of the point that divides a line segment into a specific ratio is given by:

x=(mm+n)(x2βˆ’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$ is the ratio of the part from point K to the point J.

  • n$ is the ratio of the part from point J to the point K.

  • x_1$ is the $x$-coordinate of point K.

  • x_2$ is the $x$-coordinate of point J.

Applying the Formula

In this problem, we are given that the ratio of the part from point K to the point J is $1$ and the ratio of the part from point J to the point K is $3$. Therefore, we can substitute these values into the formula:

x=(11+3)(x2βˆ’x1)+x1{ x = \left(\frac{1}{1+3}\right)\left(x_2-x_1\right)+x_1 }

Simplifying the Formula

Simplifying the formula, we get:

x=(14)(x2βˆ’x1)+x1{ x = \left(\frac{1}{4}\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 $x$-coordinates of points K and J. However, the problem does not provide this information. Therefore, we will assume that the $x$-coordinates of points K and J are $x_1 = -1$ and $x_2 = 7$, respectively.

Substituting the Values

Substituting the values of $x_1$ and $x_2$ into the formula, we get:

x=(14)(7βˆ’(βˆ’1))+(βˆ’1){ x = \left(\frac{1}{4}\right)\left(7-(-1)\right)+(-1) }

Evaluating the Expression

Evaluating the expression, we get:

x=(14)(8)+(βˆ’1){ x = \left(\frac{1}{4}\right)\left(8\right)+(-1) }

x=2+(βˆ’1){ x = 2+(-1) }

x=1{ x = 1 }

Conclusion

In this article, we explored how to find the $x$-coordinate of the point that divides the directed line segment from K to J into a ratio of $1:3$. We used the formula for dividing a line segment into a specific ratio and applied it to the given problem. We assumed that the $x$-coordinates of points K and J are $x_1 = -1$ and $x_2 = 7$, respectively. The $x$-coordinate of the point that divides the line segment is $x = 1$.

Final Answer

The final answer is: 1\boxed{1}

Introduction

In our previous article, we explored how to find the $x$-coordinate of the point that divides the directed line segment from K to J into a ratio of $1:3$. We used the formula for dividing a line segment into a specific ratio and applied it to the given problem. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the formula for dividing a line segment into a specific ratio?

A: The formula for dividing a line segment into a specific ratio is given by:

x=(mm+n)(x2βˆ’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$ is the ratio of the part from point K to the point J.

  • n$ is the ratio of the part from point J to the point K.

  • x_1$ is the $x$-coordinate of point K.

  • x_2$ is the $x$-coordinate of point J.

Q: How do I apply the formula to find the $x$-coordinate of the point that divides the line segment?

A: To apply the formula, you need to substitute the values of $m$, $n$, $x_1$, and $x_2$ into the formula. Then, simplify the expression to find the $x$-coordinate of the point that divides the line segment.

Q: What if the ratio of the part from point K to the point J is not equal to 1?

A: If the ratio of the part from point K to the point J is not equal to 1, you need to adjust the formula accordingly. For example, if the ratio is $2:3$, you would substitute $m = 2$ and $n = 3$ into the formula.

Q: Can I use the formula to find the $y$-coordinate of the point that divides the line segment?

A: Yes, you can use the formula to find the $y$-coordinate of the point that divides the line segment. Simply substitute the values of $m$, $n$, $y_1$, and $y_2$ into the formula, where $y_1$ is the $y$-coordinate of point K and $y_2$ is the $y$-coordinate of point J.

Q: What if the line segment is not horizontal or vertical?

A: If the line segment is not horizontal or vertical, you need to use the formula for the slope of a line to find the $x$-coordinate of the point that divides the line segment. The formula for the slope of a line is given by:

m=y2βˆ’y1x2βˆ’x1{ m = \frac{y_2-y_1}{x_2-x_1} }

Q: Can I use the formula to find the $x$-coordinate of the point that divides the line segment in three dimensions?

A: Yes, you can use the formula to find the $x$-coordinate of the point that divides the line segment in three dimensions. Simply substitute the values of $m$, $n$, $x_1$, $y_1$, $z_1$, $x_2$, $y_2$, and $z_2$ into the formula, where $x_1$, $y_1$, and $z_1$ are the coordinates of point K and $x_2$, $y_2$, and $z_2$ are the coordinates of point J.

Conclusion

In this article, we answered some frequently asked questions related to finding the $x$-coordinate of the point that divides the directed line segment from K to J into a ratio of $1:3$. We provided the formula for dividing a line segment into a specific ratio and explained how to apply it to find the $x$-coordinate of the point that divides the line segment. We also discussed how to use the formula to find the $y$-coordinate of the point that divides the line segment and how to handle line segments that are not horizontal or vertical.

Final Answer

The final answer is: 1\boxed{1}