What Is The \[$ Y \$\]-coordinate Of The Point That Divides The Directed Line Segment From \[$ J \$\] To \[$ K \$\] Into A Ratio Of \[$ 2:3 \$\]?$\[ Y = \left(\frac{m}{m+n}\right)\left(y_2-y_1\right) + Y_1 \\]A.

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Introduction

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 a line segment is divided into a ratio, it means that the line segment is divided 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 article, we will discuss how to find the { y $}$-coordinate of the point that divides a directed line segment from { J $}$ to { K $}$ into a ratio of { 2:3 $}$.

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. The formula is given by:

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

where { m $}$ and { n $}$ are the ratios in which the line segment is divided, and { y_1 $}$ and { y_2 $}$ are the { y $}$-coordinates of the two end points of the line segment.

Applying the Section Formula

To find the { y $}$-coordinate of the point that divides the directed line segment from { J $}$ to { K $}$ into a ratio of { 2:3 $}$, we can use the section formula. Let's assume that the { y $}$-coordinates of the two end points are { y_1 $}$ and { y_2 $}$, and the ratios are { m = 2 $}$ and { n = 3 $}$.

Substituting these values into the section formula, we get:

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

Simplifying the expression, we get:

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

Finding the { y $}$-coordinate

To find the { y $}$-coordinate of the point that divides the directed line segment from { J $}$ to { K $}$ into a ratio of { 2:3 $}$, we need to know the { y $}$-coordinates of the two end points. Let's assume that the { y $}$-coordinates of the two end points are { y_1 = 2 $}$ and { y_2 = 5 $}$.

Substituting these values into the expression, we get:

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

Simplifying the expression, we get:

y=(25)(3)+2{ y = \left(\frac{2}{5}\right)\left(3\right) + 2 }

y=65+2{ y = \frac{6}{5} + 2 }

y=65+105{ y = \frac{6}{5} + \frac{10}{5} }

y=165{ y = \frac{16}{5} }

Conclusion

In this article, we discussed how to find the { y $}$-coordinate of the point that divides a directed line segment from { J $}$ to { K $}$ into a ratio of { 2:3 $}$. We used the section formula to find the { y $}$-coordinate, and we assumed that the { y $}$-coordinates of the two end points are { y_1 = 2 $}$ and { y_2 = 5 $}$. We simplified the expression and found that the { y $}$-coordinate of the point that divides the directed line segment is { \frac{16}{5} $}$.

Example Problems

  • Find the { y $}$-coordinate of the point that divides the directed line segment from { A $}$ to { B $}$ into a ratio of { 3:4 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 1 $}$ and { y_2 = 4 $}$.
  • Find the { y $}$-coordinate of the point that divides the directed line segment from { C $}$ to { D $}$ into a ratio of { 2:5 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 3 $}$ and { y_2 = 6 $}$.

Solved Problems

  • Find the { y $}$-coordinate of the point that divides the directed line segment from { E $}$ to { F $}$ into a ratio of { 4:3 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 2 $}$ and { y_2 = 5 $}$.
  • Find the { y $}$-coordinate of the point that divides the directed line segment from { G $}$ to { H $}$ into a ratio of { 5:2 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 4 $}$ and { y_2 = 7 $}$.

Key Takeaways

  • 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 section formula is given by { y = \left(\frac{m}{m+n}\right)\left(y_2-y_1\right) + y_1 $}$, where { m $}$ and { n $}$ are the ratios in which the line segment is divided, and { y_1 $}$ and { y_2 $}$ are the { y $}$-coordinates of the two end points.
  • To find the { y $}$-coordinate of the point that divides a directed line segment into a ratio, we can use the section formula and substitute the given values into the formula.

Frequently Asked Questions

  • What is the section formula?
  • How do I find the { y $}$-coordinate of the point that divides a directed line segment into a ratio?
  • What are the ratios in which the line segment is divided?
  • What are the { y $}$-coordinates of the two end points?

Conclusion

In this article, we discussed how to find the { y $}$-coordinate of the point that divides a directed line segment from { J $}$ to { K $}$ into a ratio of { 2:3 $}$. We used the section formula to find the { y $}$-coordinate, and we assumed that the { y $}$-coordinates of the two end points are { y_1 = 2 $}$ and { y_2 = 5 $}$. We simplified the expression and found that the { y $}$-coordinate of the point that divides the directed line segment is { \frac{16}{5} $}$.

Introduction

In our previous article, we discussed how to find the { y $}$-coordinate of the point that divides a directed line segment from { J $}$ to { K $}$ into a ratio of { 2:3 $}$. We used the section formula to find the { y $}$-coordinate, and we assumed that the { y $}$-coordinates of the two end points are { y_1 = 2 $}$ and { y_2 = 5 $}$. In this article, we will answer some frequently asked questions related to finding the { y $}$-coordinate of the point that divides a directed line segment.

Q&A

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. The formula is given by { y = \left(\frac{m}{m+n}\right)\left(y_2-y_1\right) + y_1 $}$, where { m $}$ and { n $}$ are the ratios in which the line segment is divided, and { y_1 $}$ and { y_2 $}$ are the { y $}$-coordinates of the two end points.

Q: How do I find the { y $}$-coordinate of the point that divides a directed line segment into a ratio?

A: To find the { y $}$-coordinate of the point that divides a directed line segment into a ratio, you can use the section formula and substitute the given values into the formula. You will need to know the { y $}$-coordinates of the two end points and the ratios in which the line segment is divided.

Q: What are the ratios in which the line segment is divided?

A: The ratios in which the line segment is divided are given by { m $}$ and { n $}$, where { m $}$ is the ratio of the first part of the line segment and { n $}$ is the ratio of the second part of the line segment.

Q: What are the { y $}$-coordinates of the two end points?

A: The { y $}$-coordinates of the two end points are given by { y_1 $}$ and { y_2 $}$, where { y_1 $}$ is the { y $}$-coordinate of the first end point and { y_2 $}$ is the { y $}$-coordinate of the second end point.

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

A: Yes, you can use the section formula to find the { x $}$-coordinate of the point that divides a directed line segment. The formula is given by { x = \left(\frac{m}{m+n}\right)\left(x_2-x_1\right) + x_1 $}$, where { 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.

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 formula simplifies to { y = \frac{y_1 + y_2}{2} $}$, where { y_1 $}$ and { y_2 $}$ are the { y $}$-coordinates of the two end points.

Conclusion

In this article, we answered some frequently asked questions related to finding the { y $}$-coordinate of the point that divides a directed line segment. We discussed the section formula and how to use it to find the { y $}$-coordinate of the point that divides a directed line segment into a ratio. We also answered questions about the ratios in which the line segment is divided and the { y $}$-coordinates of the two end points.

Example Problems

  • Find the { y $}$-coordinate of the point that divides the directed line segment from { A $}$ to { B $}$ into a ratio of { 3:4 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 1 $}$ and { y_2 = 4 $}$.
  • Find the { y $}$-coordinate of the point that divides the directed line segment from { C $}$ to { D $}$ into a ratio of { 2:5 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 3 $}$ and { y_2 = 6 $}$.

Solved Problems

  • Find the { y $}$-coordinate of the point that divides the directed line segment from { E $}$ to { F $}$ into a ratio of { 4:3 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 2 $}$ and { y_2 = 5 $}$.
  • Find the { y $}$-coordinate of the point that divides the directed line segment from { G $}$ to { H $}$ into a ratio of { 5:2 $}$, where the { y $}$-coordinates of the two end points are { y_1 = 4 $}$ and { y_2 = 7 $}$.

Key Takeaways

  • 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 section formula is given by { y = \left(\frac{m}{m+n}\right)\left(y_2-y_1\right) + y_1 $}$, where { m $}$ and { n $}$ are the ratios in which the line segment is divided, and { y_1 $}$ and { y_2 $}$ are the { y $}$-coordinates of the two end points.
  • To find the { y $}$-coordinate of the point that divides a directed line segment into a ratio, you can use the section formula and substitute the given values into the formula.

Frequently Asked Questions

  • What is the section formula?
  • How do I find the { y $}$-coordinate of the point that divides a directed line segment into a ratio?
  • What are the ratios in which the line segment is divided?
  • What are the { y $}$-coordinates of the two end points?

Conclusion

In this article, we answered some frequently asked questions related to finding the { y $}$-coordinate of the point that divides a directed line segment. We discussed the section formula and how to use it to find the { y $}$-coordinate of the point that divides a directed line segment into a ratio. We also answered questions about the ratios in which the line segment is divided and the { y $}$-coordinates of the two end points.