Using Ratios to Find the Coordinates

Oct 13, 2025 by miodeya.com 113 views

What are the coordinates of point ${$ P $}$ on the directed line segment from ${$ P $}$ to ${$ Q $}$ such that ${$ P $}$ is ${$ \frac{5}{6}$}$ the length of the line segment from ${$ R $}$ to ${$ Q $}$?

Understanding the Problem

The problem involves finding the coordinates of a point ${$ P $}$ on a directed line segment from ${$ P $}$ to ${$ Q $}$ such that ${$ P $}$ is ${$ \frac{5}{6}$}$ the length of the line segment from ${$ R $}$ to ${$ Q $}$. To solve this problem, we need to understand the concept of directed line segments and how to find the coordinates of a point on a line segment.

Directed Line Segments

A directed line segment is a line segment with a specific direction. It is represented by two points, ${$ P $}$ and ${$ Q $}$, where ${$ P $}$ is the starting point and ${$ Q $}$ is the ending point. The direction of the line segment is from ${$ P $}$ to ${$ Q $}$.

Finding the Coordinates of Point ${$ P $}$

To find the coordinates of point ${$ P $}$, we need to use the concept of ratios. We are given that ${$ P $}$ is ${$ \frac{5}{6}$}$ the length of the line segment from ${$ R $}$ to ${$ Q $}$. This means that the distance from ${$ R $}$ to ${$ P $}$ is ${$ \frac{5}{6}$}$ of the distance from ${$ R $}$ to ${$ Q $}$.

Using Ratios to Find the Coordinates

Let's assume that the coordinates of ${$ R $}$ are ${$ (x_1, y_1) $}$ and the coordinates of ${$ Q $}$ are ${$ (x_2, y_2) $}$. We can use the concept of ratios to find the coordinates of ${$ P $}$.

The distance from ${$ R $}$ to ${$ P $}$ is ${$ \frac{5}{6}$}$ of the distance from ${$ R $}$ to ${$ Q $}$. This means that the coordinates of ${$ P $}$ can be found by multiplying the coordinates of ${$ R $}$ by ${$ \frac{5}{6}$}$ and adding the result to the coordinates of ${$ R $}$.

Calculating the Coordinates of ${$ P $}$

Let's calculate the coordinates of ${$ P $}$ using the formula:

${$ (x, y) = (x_1 + \frac{5}{6}(x_2 - x_1), y_1 + \frac{5}{6}(y_2 - y_1)) $}$

Example

Let's assume that the coordinates of ${$ R $}$ are ${$ (1, 2) $}$ and the coordinates of ${$ Q $}$ are ${$ (4, 6) $}$. We can use the formula to find the coordinates of ${$ P $}$.

${$ (x, y) = (1 + \frac{5}{6}(4 - 1), 2 + \frac{5}{6}(6 - 2)) $}$

${$ (x, y) = (1 + \frac{5}{6}(3), 2 + \frac{5}{6}(4)) $}$

${$ (x, y) = (1 + \frac{15}{6}, 2 + \frac{20}{6}) $}$

${$ (x, y) = (1 + \frac{5}{2}, 2 + \frac{10}{3}) $}$

${$ (x, y) = (\frac{7}{2}, \frac{16}{3}) $}$

Conclusion

In this article, we discussed how to find the coordinates of a point ${$ P $}$ on a directed line segment from ${$ P $}$ to ${$ Q $}$ such that ${$ P $}$ is ${$ \frac{5}{6}$}$ the length of the line segment from ${$ R $}$ to ${$ Q $}$. We used the concept of ratios to find the coordinates of ${$ P $}$ and provided an example to illustrate the calculation.

Key Takeaways

A directed line segment is a line segment with a specific direction.
The coordinates of a point on a line segment can be found using the concept of ratios.
The formula for finding the coordinates of a point on a line segment is ${$ (x, y) = (x_1 + \frac{m}{n}(x_2 - x_1), y_1 + \frac{m}{n}(y_2 - y_1)) $}$, where ${$ m $}$ and ${$ n $}$ are the ratio and the total number of parts, respectively.

Further Reading

For more information on directed line segments and how to find the coordinates of a point on a line segment, please refer to the following resources:

Directed Line Segments
Ratios and Proportions
Coordinate Geometry
Q&A: Directed Line Segments and Finding Coordinates

Q: What is a directed line segment?

A: A directed line segment is a line segment with a specific direction. It is represented by two points, ${$ P $}$ and ${$ Q $}$, where ${$ P $}$ is the starting point and ${$ Q $}$ is the ending point. The direction of the line segment is from ${$ P $}$ to ${$ Q $}$.

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

A: To find the coordinates of a point on a directed line segment, you can use the concept of ratios. You need to know the coordinates of the starting point and the ending point of the line segment, and then use the ratio to find the coordinates of the point.

Q: What is the formula for finding the coordinates of a point on a line segment?

A: The formula for finding the coordinates of a point on a line segment is ${$ (x, y) = (x_1 + \frac{m}{n}(x_2 - x_1), y_1 + \frac{m}{n}(y_2 - y_1)) $}$, where ${$ m $}$ and ${$ n $}$ are the ratio and the total number of parts, respectively.

Q: How do I use the formula to find the coordinates of a point on a line segment?

A: To use the formula, you need to know the coordinates of the starting point and the ending point of the line segment, and then plug in the values into the formula. For example, if the coordinates of the starting point are ${$ (1, 2) $}$ and the coordinates of the ending point are ${$ (4, 6) $}$, and you want to find the coordinates of a point that is ${$ \frac{5}{6}$}$ of the way from the starting point to the ending point, you would use the formula as follows:

${$ (x, y) = (1 + \frac{5}{6}(4 - 1), 2 + \frac{5}{6}(6 - 2)) $}$

Q: What if I don't know the coordinates of the starting point and the ending point of the line segment?

A: If you don't know the coordinates of the starting point and the ending point of the line segment, you can use other methods to find the coordinates of a point on the line segment. For example, you can use the concept of slope to find the equation of the line segment, and then use the equation to find the coordinates of a point on the line segment.

Q: Can I use the formula to find the coordinates of a point on a line segment that is not directed?

A: No, the formula is specifically designed for finding the coordinates of a point on a directed line segment. If you are working with a line segment that is not directed, you will need to use a different method to find the coordinates of a point on the line segment.

Q: Are there any other formulas or methods for finding the coordinates of a point on a line segment?

A: Yes, there are other formulas and methods for finding the coordinates of a point on a line segment. For example, you can use the concept of parametric equations to find the coordinates of a point on a line segment. You can also use the concept of vector equations to find the coordinates of a point on a line segment.

Q: Can I use the formula to find the coordinates of a point on a line segment that is not a straight line?

A: No, the formula is specifically designed for finding the coordinates of a point on a straight line. If you are working with a line segment that is not a straight line, you will need to use a different method to find the coordinates of a point on the line segment.

Q: Are there any online resources or tools that can help me find the coordinates of a point on a line segment?

A: Yes, there are many online resources and tools that can help you find the coordinates of a point on a line segment. For example, you can use online graphing calculators or geometry software to find the coordinates of a point on a line segment.

Q: Can I use the formula to find the coordinates of a point on a line segment that is in 3D space?

A: No, the formula is specifically designed for finding the coordinates of a point on a line segment in 2D space. If you are working with a line segment in 3D space, you will need to use a different method to find the coordinates of a point on the line segment.

Q: Are there any books or textbooks that can help me learn more about directed line segments and finding coordinates?

A: Yes, there are many books and textbooks that can help you learn more about directed line segments and finding coordinates. For example, you can check out textbooks on geometry, trigonometry, or calculus to learn more about directed line segments and finding coordinates.