Understanding the Concept of a Directed Line Segment

Oct 13, 2025 by wbrum.org 52 views

What is the $x$ -coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$ ?

Understanding the Concept of a Directed Line Segment

A directed line segment is a line segment with a specific direction, often represented by an arrow. In this context, we are dealing with a line segment from point J to point K. The directed line segment can be thought of as a vector, with the starting point being J and the ending point being K.

The Concept of a Ratio in a Directed Line Segment

When we divide a directed line segment into a ratio, we are essentially dividing the line segment into two parts, with the first part being a certain fraction of the total length and the second part being the remaining fraction. In this case, we are dividing the line segment from J to K into a ratio of 2:5. This means that the first part of the line segment (from J to the point of division) is 2 units long, and the second part (from the point of division to K) is 5 units long.

The Formula for Finding the $x$ -coordinate of the Point of Division

The formula for finding the $x$ -coordinate of the point that divides a directed line segment into a ratio is given by:

$x = \left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$

where $x$ is the $x$ -coordinate of the point of division, $m$ and $n$ are the lengths of the two parts of the line segment, $x_1$ and $x_2$ are the $x$ -coordinates of the starting and ending points of the line segment, respectively.

Applying the Formula to the Given Problem

In this problem, we are given that the directed line segment from J to K is divided into a ratio of 2:5. We are also given that the $x$ -coordinates of points J and K are $x_1$ and $x_2$ , respectively. We need to find the $x$ -coordinate of the point that divides the line segment into this ratio.

Using the formula, we can substitute the values of $m$ , $n$ , $x_1$ , and $x_2$ into the equation:

$x = \left(\frac{2}{2+5}\right)\left(x_2-x_1\right)+x_1$

Simplifying the equation, we get:

$x = \left(\frac{2}{7}\right)\left(x_2-x_1\right)+x_1$

Solving for the $x$ -coordinate of the Point of Division

To solve for the $x$ -coordinate of the point of division, we need to simplify the equation further. We can start by multiplying the fraction $\frac{2}{7}$ by the expression $(x_2-x_1)$ :

$x = \frac{2}{7}(x_2-x_1)+x_1$

Next, we can distribute the fraction to the terms inside the parentheses:

$x = \frac{2x_2}{7} - \frac{2x_1}{7} + x_1$

Now, we can combine like terms:

$x = \frac{2x_2}{7} + \left(-\frac{2x_1}{7} + x_1\right)$

To simplify the expression inside the parentheses, we can multiply the fraction $-\frac{2}{7}$ by $x_1$ :

$x = \frac{2x_2}{7} + \left(-\frac{2x_1}{7} + \frac{7x_1}{7}\right)$

Now, we can combine like terms:

$x = \frac{2x_2}{7} + \frac{5x_1}{7}$

Finding the Final Answer

To find the final answer, we need to simplify the expression further. We can start by combining the fractions:

$x = \frac{2x_2 + 5x_1}{7}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x = \frac{2x_2 + 5x_1}{7}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the
Q&A: What is the $x$ -coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$ ?

Q: What is the formula for finding the $x$ -coordinate of the point that divides a directed line segment into a ratio?

A: The formula for finding the $x$ -coordinate of the point that divides a directed line segment into a ratio is given by:

$x = \left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$

Q: How do I apply the formula to find the $x$ -coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$ ?

A: To apply the formula, you need to substitute the values of $m$ , $n$ , $x_1$ , and $x_2$ into the equation. In this case, $m = 2$ , $n = 5$ , $x_1$ is the $x$ -coordinate of point J, and $x_2$ is the $x$ -coordinate of point K.

Q: What is the $x$ -coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$ ?

A: To find the $x$ -coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$ , you need to substitute the values of $m$ , $n$ , $x_1$ , and $x_2$ into the equation and simplify the expression.

Q: How do I simplify the expression to find the $x$ -coordinate of the point that divides the directed line segment from J to K into a ratio of $2: 5$ ?

A: To simplify the expression, you need to combine like terms and divide both the numerator and the denominator by their greatest common divisor, which is 1.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

$x = \frac{2x_2 + 5x_1}{7}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Q: What is the value of $x_2$ ?

A: The value of $x_2$ is:

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Q: How do I find the value of $x_1$ ?

A: To find the value of $x_1$ , you need to substitute the values of $x_2$ and $x_1$ into the equation and solve for $x_1$ .

Q: What is the value of $x_1$ ?

A: The value of $x_1$ is:

$x_1 = \frac{2x_2 + 5x_1}{7}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$

Q: What is the value of $x_1$ ?

A: The value of $x_1$ is:

$x_1 = \frac{2x_2 + 5x_1}{7}$

Since we are given that the ratio of the line segment is 2:5, we can set up the equation:

$\frac{2x_2 + 5x_1}{7} = \frac{2}{5}$

To solve for $x_2$ , we can multiply both sides of the equation by 7:

$2x_2 + 5x_1 = \frac{14}{5}$

Next, we can multiply both sides of the equation by 5 to eliminate the fraction:

$10x_2 + 25x_1 = 14$

Now, we can solve for $x_2$ by subtracting $25x_1$ from both sides of the equation:

$10x_2 = 14 - 25x_1$

Next, we can divide both sides of the equation by 10:

$x_2 = \frac{14 - 25x_1}{10}$

Now, we can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$x_2 = \frac{14}{10} - \frac{25x_1}{10}$

$x_2 = \frac{7}{5} - \frac{5x_1}{2}$