What Is The \[$x\$\]-coordinate Of The Point That Divides The Directed Line Segment From \[$K\$\] To \[$J\$\] In The Ratio \[$m:n\$\]?$\[ x = \left(\frac{m}{m+n}\right)(x_2-x_1) + X_1 \\]A. \[$-1\$\] B.

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What is the {x$}$-coordinate of the point that divides the directed line segment from {K$}$ to {J$}$ in the ratio {m:n$}$

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, often represented by an arrow. When we divide a directed line segment into a ratio, we are essentially finding a point that divides the segment into two parts, with the first part being a fraction of the total length. 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$}$ in the ratio {m:n$}$.

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

x=(mm+n)(x2−x1)+x1x = \left(\frac{m}{m+n}\right)(x_2-x_1) + 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 two end points of the line segment.

Understanding the Formula

To understand the formula, let's break it down into its components. The first term, {\left(\frac{m}{m+n}\right)(x_2-x_1)$}$, represents the fraction of the total length of the line segment that the point divides. The second term, {x_1$}$, represents the {x$}$-coordinate of the first end point of the line segment.

Applying the Formula

To apply the formula, we need to know the {x$}$-coordinates of the two end points of the line segment, as well as the ratios in which the line segment is divided. Let's consider an example to illustrate this.

Example

Suppose we have a line segment with end points {J$}$ and {K$}$, with {x$}$-coordinates {x_1 = -1$}$ and {x_2 = 3$}$, respectively. We want to find the {x$}$-coordinate of the point that divides the line segment in the ratio $$23$$.

Using the section formula, we can plug in the values as follows:

x=(22+3)(3−(−1))+(−1)x = \left(\frac{2}{2+3}\right)(3-(-1)) + (-1)

Simplifying the expression, we get:

x=(25)(4)+(−1)x = \left(\frac{2}{5}\right)(4) + (-1)

x=85−1x = \frac{8}{5} - 1

x=35x = \frac{3}{5}

Therefore, the {x$}$-coordinate of the point that divides the line segment in the ratio $$23$$ is {\frac{3}{5}$}$.

In conclusion, the section formula is a powerful tool for finding the coordinates of a point that divides a line segment into a given ratio. By understanding the formula and applying it to specific problems, we can find the {x$}$-coordinate of the point that divides the directed line segment from {K$}$ to {J$}$ in the ratio {m:n$}$. Whether you're a student or a professional, the section formula is an essential tool to have in your mathematical toolkit.

  • What is the section formula? The section formula is a mathematical formula used to find the coordinates of a point that divides a line segment into a given ratio.
  • How do I apply the section formula? To apply the section formula, you need to know the {x$}$-coordinates of the two end points of the line segment, as well as the ratios in which the line segment is divided.
  • What is the {x$}$-coordinate of the point that divides the line segment in the ratio $$23$$? The {x$}$-coordinate of the point that divides the line segment in the ratio $$23$$ is {\frac{3}{5}$}$.

Q: What is the section formula?

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

Q: How do I apply the section formula?

A: To apply the section formula, you need to know the {x$}$-coordinates of the two end points of the line segment, as well as the ratios in which the line segment is divided.

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

A: The formula for finding the {x$}$-coordinate of a point that divides a line segment is:

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

Q: What are the variables in the section formula?

A: The variables in the section formula are:

  • {m$}$ and {n$}$: the ratios in which the line segment is divided
  • {x_1$}$ and {x_2$}$: the {x$}$-coordinates of the two end points of the line segment
  • {x$}$: the {x$}$-coordinate of the point that divides the line segment

Q: How do I find the {x$}$-coordinate of the point that divides a line segment in a given ratio?

A: To find the {x$}$-coordinate of the point that divides a line segment in a given ratio, you can use the section formula. Simply plug in the values of the ratios and the {x$}$-coordinates of the end points into the formula.

Q: What is the significance of the section formula in coordinate geometry?

A: The section formula is a fundamental concept in coordinate geometry, as it allows us to find the coordinates of a point that divides a line segment into a given ratio. This is a crucial concept in many areas of mathematics and science, including geometry, trigonometry, and calculus.

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

A: Yes, you can use the section formula to find the {y$}$-coordinate of a point that divides a line segment. The formula for finding the {y$}$-coordinate is:

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

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

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

  • Architecture: finding the coordinates of a point that divides a line segment is essential in architecture, where it is used to design buildings and other structures.
  • Engineering: the section formula is used in engineering to find the coordinates of a point that divides a line segment, which is essential in designing bridges, roads, and other infrastructure.
  • Computer Graphics: the section formula is used in computer graphics to find the coordinates of a point that divides a line segment, which is essential in creating 3D models and animations.
  • Not using the correct formula for finding the {x$}$-coordinate of a point that divides a line segment.
  • Not plugging in the correct values into the formula.
  • Not simplifying the expression after plugging in the values.
  • Not checking the units of the answer.
  • Make sure to use the correct formula for finding the {x$}$-coordinate of a point that divides a line segment.
  • Plug in the correct values into the formula.
  • Simplify the expression after plugging in the values.
  • Check the units of the answer.

In conclusion, the section formula is a powerful tool for finding the coordinates of a point that divides a line segment into a given ratio. By understanding the formula and applying it to specific problems, we can find the {x$}$-coordinate of the point that divides the directed line segment from {K$}$ to {J$}$ in the ratio {m:n$}$. Whether you're a student or a professional, the section formula is an essential tool to have in your mathematical toolkit.