Point { A $}$ Is Located At { (4,1) $}$ And Point { B $}$ Is Located At { (9,13) $}$.What Are The Coordinates Of The Point That Partitions The Directed Line Segment { \overline{AB}$}$ In A [$

Introduction

In geometry, a directed line segment is a line segment with a specific direction. When we partition a directed line segment, we are essentially dividing it into two equal parts. In this article, we will explore how to find the coordinates of the point that partitions the directed line segment ${\overline{AB}}$ in a given ratio.

Understanding the Problem

We are given two points, A and B, with coordinates (4,1) and (9,13) respectively. We need to find the coordinates of the point that partitions the directed line segment ${\overline{AB}}$ in a given ratio.

The Partitioning Formula

To find the coordinates of the point that partitions the directed line segment ${\overline{AB}}$ in a given ratio, we can use the following formula:

${(x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)}$

where (x1, y1) and (x2, y2) are the coordinates of points A and B respectively, and m:n is the ratio in which the line segment is partitioned.

Applying the Formula

In this case, we want to partition the directed line segment ${\overline{AB}}$ in a 1:2 ratio. This means that the point we are looking for is 1/3 of the way from point A to point B.

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

${(x, y) = \left(\frac{1(9) + 2(4)}{1 + 2}, \frac{1(13) + 2(1)}{1 + 2}\right)}$

Simplifying the expression, we get:

${(x, y) = \left(\frac{9 + 8}{3}, \frac{13 + 2}{3}\right)}$

${(x, y) = \left(\frac{17}{3}, \frac{15}{3}\right)}$

${(x, y) = \left(5.67, 5\right)}$

Conclusion

In this article, we have explored how to find the coordinates of the point that partitions the directed line segment ${\overline{AB}}$ in a given ratio. We have used the partitioning formula to find the coordinates of the point that partitions the directed line segment ${\overline{AB}}$ in a 1:2 ratio.

Example Use Cases

1. Architecture: When designing buildings, architects need to partition directed line segments to create symmetrical structures.
2. Engineering: Engineers use partitioning to divide directed line segments when designing mechanical systems, such as gears and pulleys.
3. Computer Graphics: In computer graphics, partitioning is used to create 3D models and animations.

Tips and Tricks

1. Use the Partitioning Formula: The partitioning formula is a powerful tool for finding the coordinates of the point that partitions a directed line segment.
2. Choose the Right Ratio: The ratio in which the line segment is partitioned will affect the coordinates of the point.
3. Simplify the Expression: Simplify the expression to get the final coordinates of the point.

Common Mistakes

1. Incorrect Ratio: Using the wrong ratio will result in incorrect coordinates.
2. Incorrect Formula: Using the wrong formula will result in incorrect coordinates.
3. Not Simplifying the Expression: Failing to simplify the expression will result in incorrect coordinates.

Conclusion

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Frequently Asked Questions

Q: What is point partitioning?

A: Point partitioning is the process of dividing a directed line segment into two equal parts. This is done by finding the coordinates of the point that partitions the line segment in a given ratio.

Q: Why is point partitioning important?

A: Point partitioning is important in various fields such as architecture, engineering, and computer graphics. It is used to create symmetrical structures, divide directed line segments, and create 3D models and animations.

Q: How do I find the coordinates of the point that partitions a directed line segment?

A: To find the coordinates of the point that partitions a directed line segment, you can use the partitioning formula:

${(x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)}$

where (x1, y1) and (x2, y2) are the coordinates of points A and B respectively, and m:n is the ratio in which the line segment is partitioned.

Q: What is the ratio in which the line segment is partitioned?

A: The ratio in which the line segment is partitioned is the proportion of the line segment that the point divides. For example, if the ratio is 1:2, the point divides the line segment into two parts, with the first part being 1/3 of the total length and the second part being 2/3 of the total length.

Q: How do I choose the right ratio?

A: The right ratio depends on the specific problem you are trying to solve. You need to choose a ratio that makes sense in the context of the problem. For example, if you are designing a building, you may want to choose a ratio that creates a symmetrical structure.

Q: What are some common mistakes to avoid when partitioning a directed line segment?

A: Some common mistakes to avoid when partitioning a directed line segment include:

• Using the wrong ratio
• Using the wrong formula
• Not simplifying the expression
• Not checking the units of the coordinates

Q: How do I simplify the expression?

A: To simplify the expression, you need to combine like terms and cancel out any common factors. For example, if you have the expression (x, y) = (17/3, 15/3), you can simplify it to (x, y) = (5.67, 5).

Q: What are some real-world applications of point partitioning?

A: Some real-world applications of point partitioning include:

• Architecture: Point partitioning is used to create symmetrical structures and divide directed line segments.
• Engineering: Point partitioning is used to divide directed line segments and create 3D models and animations.
• Computer Graphics: Point partitioning is used to create 3D models and animations.

Q: How do I practice point partitioning?

A: To practice point partitioning, you can try solving problems that involve partitioning directed line segments. You can also use online resources and practice exercises to help you improve your skills.

Q: What are some resources for learning more about point partitioning?

A: Some resources for learning more about point partitioning include:

• Online tutorials and videos
• Practice exercises and worksheets
• Textbooks and online courses
• Online communities and forums

Conclusion

In conclusion, point partitioning is an important concept in geometry that has many real-world applications. By understanding the partitioning formula and how to choose the right ratio, you can solve problems in mathematics and other fields. With practice and experience, you will become proficient in using the partitioning formula to solve problems.