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 direction, which is indicated by an arrowhead. The direction of the line segment is from the starting point to the ending point. When we partition a line segment into a ratio, we are dividing the segment into two parts, with the first part being a fraction of the total length of the segment.

Formula for Partitioning a Line Segment

The formula for partitioning a line segment into a ratio of m:n is given by:

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

where $(x_1, y_1)$ is the starting point, $(x_2, y_2)$ is the ending point, and m:n is the ratio in which the segment is partitioned.

Applying the Formula

In our problem, the starting point is $(-3,4)$, the ending point is $(3,-8)$, and the ratio is 1:3. We can plug these values into the formula to find the coordinates of the point that partitions the segment into a ratio of 1 to 3.

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

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

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

(βˆ’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 our problem to find the coordinates of the point. 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 many practical applications in mathematics and real-life situations. Here are a few example use cases:

  • Architecture: When designing a building, architects may need to partition a line segment to create a specific ratio of windows to walls.
  • Engineering: Engineers may need to partition a line segment to create a specific ratio of pipes to valves in a plumbing system.
  • Art: Artists may need to partition a line segment to create a specific ratio of colors in a painting.

Tips and Tricks

Here are a few tips and tricks to help you solve problems involving partitioning a line segment:

  • Use the formula: The formula for partitioning a line segment into a given ratio is a powerful tool that can help you solve problems quickly and efficiently.
  • Plug in the values: Make sure to plug in the correct values into the formula, including the starting point, ending point, and ratio.
  • Simplify the expression: Simplify the expression to find the coordinates of the point.

Conclusion

In conclusion, partitioning a line segment into a given ratio is a fundamental concept in mathematics that has many practical applications. By using the formula and applying it to our problem, we have found 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 hope that this article has provided you with a better understanding of this concept and how to apply it to solve problems.

Introduction

In our previous article, we 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. In this article, we will answer some frequently asked questions about partitioning a line segment into a given ratio.

Q1: What is the formula for partitioning a line segment into a given ratio?

A1: 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 $(x_1, y_1)$ is the starting point, $(x_2, y_2)$ is the ending point, and m:n is the ratio in which the segment is partitioned.

Q2: How do I apply the formula to find the coordinates of the point?

A2: To apply the formula, you need to plug in the correct values into the formula, including the starting point, ending point, and ratio. Then, simplify the expression to find the coordinates of the point.

Q3: What if the ratio is not in the simplest form?

A3: If the ratio is not in the simplest form, you need to simplify it before applying the formula. For example, if the ratio is 2:6, you can simplify it to 1:3 by dividing both numbers by 2.

Q4: Can I partition a line segment into a ratio of 0:1 or 1:0?

A4: No, you cannot partition a line segment into a ratio of 0:1 or 1:0. These ratios are not valid, as they would result in a line segment with zero length.

Q5: How do I find the coordinates of the point on a line segment that partitions the segment into a ratio of 2:3?

A5: To find the coordinates of the point on a line segment that partitions the segment into a ratio of 2:3, you can use the formula:

(2x2+3x12+3,2y2+3y12+3)\left( \frac{2x_2 + 3x_1}{2+3}, \frac{2y_2 + 3y_1}{2+3} \right)

where $(x_1, y_1)$ is the starting point, $(x_2, y_2)$ is the ending point.

Q6: Can I use the formula to find the coordinates of the point on a line segment that partitions the segment into a ratio of 3:2?

A6: Yes, you can use the formula to find the coordinates of the point on a line segment that partitions the segment into a ratio of 3:2. The formula remains the same:

(3x2+2x13+2,3y2+2y13+2)\left( \frac{3x_2 + 2x_1}{3+2}, \frac{3y_2 + 2y_1}{3+2} \right)

Q7: How do I find the coordinates of the point on a line segment that partitions the segment into a ratio of 1:1?

A7: To find the coordinates of the point on a line segment that partitions the segment into a ratio of 1:1, you can use the formula:

(x2+x11+1,y2+y11+1)\left( \frac{x_2 + x_1}{1+1}, \frac{y_2 + y_1}{1+1} \right)

(x2+x12,y2+y12)\left( \frac{x_2 + x_1}{2}, \frac{y_2 + y_1}{2} \right)

Conclusion

In this article, we have answered some frequently asked questions about partitioning a line segment into a given ratio. We hope that this article has provided you with a better understanding of this concept and how to apply it to solve problems. If you have any more questions, feel free to ask!

Example Use Cases

The concept of partitioning a line segment into a given ratio has many practical applications in mathematics and real-life situations. Here are a few example use cases:

  • Architecture: When designing a building, architects may need to partition a line segment to create a specific ratio of windows to walls.
  • Engineering: Engineers may need to partition a line segment to create a specific ratio of pipes to valves in a plumbing system.
  • Art: Artists may need to partition a line segment to create a specific ratio of colors in a painting.

Tips and Tricks

Here are a few tips and tricks to help you solve problems involving partitioning a line segment:

  • Use the formula: The formula for partitioning a line segment into a given ratio is a powerful tool that can help you solve problems quickly and efficiently.
  • Plug in the values: Make sure to plug in the correct values into the formula, including the starting point, ending point, and ratio.
  • Simplify the expression: Simplify the expression to find the coordinates of the point.

Conclusion

In conclusion, partitioning a line segment into a given ratio is a fundamental concept in mathematics that has many practical applications. By using the formula and applying it to our problems, we have found 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 hope that this article has provided you with a better understanding of this concept and how to apply it to solve problems.