What Are The Coordinates Of The Point On The Directed Line Segment From $(-3,4)$ To $(3,-8)$ That Partitions The Segment Into A Ratio Of 1 To 3?

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Introduction

In mathematics, particularly in geometry and coordinate geometry, we often encounter problems involving line segments and their partitions. One such problem is finding the coordinates of a point on a directed line segment that partitions the segment into a given ratio. In this article, we will explore how to find the coordinates of the point on the directed line segment from $(-3,4)$ to $(3,-8)$ that partitions the segment into a ratio of 1 to 3.

Understanding the Problem

To solve this problem, we need to understand the concept of a directed line segment and how to partition it into a given ratio. A directed line segment is a line segment with a specific direction, represented by an arrow. The ratio of partitioning a line segment refers to the proportion of the segment that is divided into two parts. In this case, we want to partition the line segment from $(-3,4)$ to $(3,-8)$ into a ratio of 1 to 3.

Formula for Partitioning a Line Segment

The formula for partitioning a line segment into a given ratio is based on the concept of similar triangles. Let's consider two points on the line segment, $(x_1,y_1)$ and $(x_2,y_2)$, and a ratio of partitioning $m:n$. The formula for finding the coordinates of the point that partitions the segment into the given ratio is:

(mx2+nx1m+n,my2+ny1m+n)\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)

Applying the Formula

Now, let's apply the formula to find the coordinates of the point on the directed line segment from $(-3,4)$ to $(3,-8)$ that partitions the segment into a ratio of 1 to 3. We have the following values:

x1=−3x_1 = -3

y1=4y_1 = 4

x2=3x_2 = 3

y2=−8y_2 = -8

m=1m = 1

n=3n = 3

Substituting these values into the formula, we get:

(1(3)+3(−3)1+3,1(−8)+3(4)1+3)\left(\frac{1(3)+3(-3)}{1+3},\frac{1(-8)+3(4)}{1+3}\right)

Calculating the Coordinates

Now, let's calculate the coordinates of the point:

(3−94,−8+124)\left(\frac{3-9}{4},\frac{-8+12}{4}\right)

(−64,44)\left(\frac{-6}{4},\frac{4}{4}\right)

(−32,1)\left(-\frac{3}{2},1\right)

Conclusion

In this article, we have explored how to find the coordinates of the point on the directed line segment from $(-3,4)$ to $(3,-8)$ that partitions the segment into a ratio of 1 to 3. We have used the formula for partitioning a line segment into a given ratio and applied it to the given problem. The coordinates of the point are $\left(-\frac{3}{2},1\right)$.

Example Use Cases

The concept of partitioning a line segment into a given ratio has numerous applications in mathematics, science, and engineering. Here are a few example use cases:

  • Geometry and Trigonometry: Partitioning a line segment is a fundamental concept in geometry and trigonometry. It is used to solve problems involving similar triangles, right triangles, and other geometric shapes.
  • Physics and Engineering: In physics and engineering, partitioning a line segment is used to model real-world problems involving motion, forces, and energies.
  • Computer Graphics: In computer graphics, partitioning a line segment is used to create 2D and 3D models, animations, and special effects.

Final Thoughts

In conclusion, finding the coordinates of the point on the directed line segment from $(-3,4)$ to $(3,-8)$ that partitions the segment into a ratio of 1 to 3 is a simple yet powerful problem that has numerous applications in mathematics, science, and engineering. By understanding the concept of partitioning a line segment and applying the formula, we can solve a wide range of problems involving similar triangles, right triangles, and other geometric shapes.

References

  • [1] "Coordinate Geometry" by Khan Academy
  • [2] "Similar Triangles" by Math Open Reference
  • [3] "Partitioning a Line Segment" by Wolfram MathWorld

Related Topics

  • Similar Triangles: Similar triangles are triangles that have the same shape but not necessarily the same size.
  • Right Triangles: Right triangles are triangles with one right angle (90 degrees).
  • Geometry and Trigonometry: Geometry and trigonometry are branches of mathematics that deal with shapes, sizes, and positions of objects.
  • Physics and Engineering: Physics and engineering are branches of science that deal with the study of the natural world and the application of scientific principles to solve real-world problems.

Q&A: Partitioning a Line Segment

Q: What is the formula for partitioning a line segment into a given ratio?

A: The formula for partitioning a line segment into a given ratio is:

(mx2+nx1m+n,my2+ny1m+n)\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)

where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points on the line segment, and $m:n$ is the ratio of partitioning.

Q: How do I apply the formula to find the coordinates of the point on the directed line segment from $(-3,4)$ to $(3,-8)$ that partitions the segment into a ratio of 1 to 3?

A: To apply the formula, substitute the values of $(x_1,y_1) = (-3,4)$, $(x_2,y_2) = (3,-8)$, $m = 1$, and $n = 3$ into the formula:

(1(3)+3(−3)1+3,1(−8)+3(4)1+3)\left(\frac{1(3)+3(-3)}{1+3},\frac{1(-8)+3(4)}{1+3}\right)

Q: What are the coordinates of the point on the directed line segment from $(-3,4)$ to $(3,-8)$ that partitions the segment into a ratio of 1 to 3?

A: The coordinates of the point are $\left(-\frac{3}{2},1\right)$.

Q: What are some example use cases of partitioning a line segment?

A: Partitioning a line segment has numerous applications in mathematics, science, and engineering, including:

  • Geometry and Trigonometry: Partitioning a line segment is a fundamental concept in geometry and trigonometry. It is used to solve problems involving similar triangles, right triangles, and other geometric shapes.
  • Physics and Engineering: In physics and engineering, partitioning a line segment is used to model real-world problems involving motion, forces, and energies.
  • Computer Graphics: In computer graphics, partitioning a line segment is used to create 2D and 3D models, animations, and special effects.

Q: What are some related topics to partitioning a line segment?

A: Some related topics to partitioning a line segment include:

  • Similar Triangles: Similar triangles are triangles that have the same shape but not necessarily the same size.
  • Right Triangles: Right triangles are triangles with one right angle (90 degrees).
  • Geometry and Trigonometry: Geometry and trigonometry are branches of mathematics that deal with shapes, sizes, and positions of objects.
  • Physics and Engineering: Physics and engineering are branches of science that deal with the study of the natural world and the application of scientific principles to solve real-world problems.

Q: Where can I find more information on partitioning a line segment?

A: You can find more information on partitioning a line segment in the following resources:

  • [1] "Coordinate Geometry" by Khan Academy
  • [2] "Similar Triangles" by Math Open Reference
  • [3] "Partitioning a Line Segment" by Wolfram MathWorld

Conclusion

In this Q&A article, we have explored the concept of partitioning a line segment into a given ratio and provided answers to some common questions. We have also discussed some example use cases and related topics to partitioning a line segment. If you have any further questions or need more information, please don't hesitate to ask.