What Are The Coordinates Of Point $P$ On The Directed Line Segment From $R$ To $Q$ Such That $P$ Is $\frac{5}{6}$ The Length Of The Line Segment From $R$ To $Q$? Round To The

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Introduction

In mathematics, particularly in geometry and coordinate geometry, we often encounter problems involving points on a line segment. These problems can be solved using various techniques, including the concept of ratios and proportions. In this article, we will explore how to find the coordinates of a point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$.

Understanding the Problem

To solve this problem, we need to understand the concept of ratios and proportions. A ratio is a comparison of two numbers, and a proportion is a statement that two ratios are equal. In this case, we are given that point $P$ 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 and Proportions

To find the coordinates of point $P$, we can use the concept of ratios and proportions. Let's assume that the coordinates of point $R$ are $(x_1, y_1)$ and the coordinates of point $Q$ are $(x_2, y_2)$. We can then use the ratio $\frac{5}{6}$ to find the coordinates of point $P$.

Calculating the Coordinates of Point $P$

Using the concept of ratios and proportions, we can calculate the coordinates of point $P$ as follows:

x2−x1x2−x1=56\frac{x_2 - x_1}{x_2 - x_1} = \frac{5}{6}

y2−y1y2−y1=56\frac{y_2 - y_1}{y_2 - y_1} = \frac{5}{6}

Simplifying the above equations, we get:

x2−x1=56(x2−x1)x_2 - x_1 = \frac{5}{6}(x_2 - x_1)

y2−y1=56(y2−y1)y_2 - y_1 = \frac{5}{6}(y_2 - y_1)

Solving for the Coordinates of Point $P$

Now that we have the equations, we can solve for the coordinates of point $P$. Let's assume that the coordinates of point $R$ are $(x_1, y_1)$ and the coordinates of point $Q$ are $(x_2, y_2)$. We can then use the equations above to find the coordinates of point $P$.

Finding the Distance from $R$ to $P$

To find the distance from $R$ to $P$, we can use the concept of ratios and proportions. Let's assume that the distance from $R$ to $P$ is $d_1$ and the distance from $R$ to $Q$ is $d_2$. We can then use the ratio $\frac{5}{6}$ to find the distance from $R$ to $P$.

Calculating the Distance from $R$ to $P$

Using the concept of ratios and proportions, we can calculate the distance from $R$ to $P$ as follows:

d1d2=56\frac{d_1}{d_2} = \frac{5}{6}

Simplifying the above equation, we get:

d1=56d2d_1 = \frac{5}{6}d_2

Solving for the Distance from $R$ to $P$

Now that we have the equation, we can solve for the distance from $R$ to $P$. Let's assume that the distance from $R$ to $Q$ is $d_2$. We can then use the equation above to find the distance from $R$ to $P$.

Conclusion

In this article, we explored how to find the coordinates of a point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$. We used the concept of ratios and proportions to solve for the coordinates of point $P$ and the distance from $R$ to $P$. This problem is a classic example of how to use ratios and proportions to solve problems in geometry and coordinate geometry.

Example

Let's consider an example to illustrate the concept. Suppose we have a line segment from $R(1, 2)$ to $Q(4, 6)$. We want to find the coordinates of point $P$ such that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$.

Step 1: Find the Distance from $R$ to $Q$

To find the distance from $R$ to $Q$, we can use the distance formula:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Plugging in the values, we get:

d=(4−1)2+(6−2)2d = \sqrt{(4 - 1)^2 + (6 - 2)^2}

d=32+42d = \sqrt{3^2 + 4^2}

d=9+16d = \sqrt{9 + 16}

d=25d = \sqrt{25}

d=5d = 5

Step 2: Find the Distance from $R$ to $P$

Using the ratio $\frac{5}{6}$, we can find the distance from $R$ to $P$:

d1=56d2d_1 = \frac{5}{6}d_2

d1=56(5)d_1 = \frac{5}{6}(5)

d1=256d_1 = \frac{25}{6}

Step 3: Find the Coordinates of Point $P$

To find the coordinates of point $P$, we can use the concept of ratios and proportions. Let's assume that the coordinates of point $P$ are $(x, y)$. We can then use the ratio $\frac{5}{6}$ to find the coordinates of point $P$.

Step 4: Solve for the Coordinates of Point $P$

Using the concept of ratios and proportions, we can solve for the coordinates of point $P$ as follows:

x−x1x2−x1=56\frac{x - x_1}{x_2 - x_1} = \frac{5}{6}

y−y1y2−y1=56\frac{y - y_1}{y_2 - y_1} = \frac{5}{6}

Simplifying the above equations, we get:

x−x1=56(x2−x1)x - x_1 = \frac{5}{6}(x_2 - x_1)

y−y1=56(y2−y1)y - y_1 = \frac{5}{6}(y_2 - y_1)

Step 5: Find the Coordinates of Point $P$

Now that we have the equations, we can solve for the coordinates of point $P$. Plugging in the values, we get:

x−1=56(4−1)x - 1 = \frac{5}{6}(4 - 1)

x−1=56(3)x - 1 = \frac{5}{6}(3)

x−1=156x - 1 = \frac{15}{6}

x−1=52x - 1 = \frac{5}{2}

x=52+1x = \frac{5}{2} + 1

x=52+22x = \frac{5}{2} + \frac{2}{2}

x=72x = \frac{7}{2}

y−2=56(6−2)y - 2 = \frac{5}{6}(6 - 2)

y−2=56(4)y - 2 = \frac{5}{6}(4)

y−2=206y - 2 = \frac{20}{6}

y−2=103y - 2 = \frac{10}{3}

y=103+2y = \frac{10}{3} + 2

y=103+63y = \frac{10}{3} + \frac{6}{3}

y=163y = \frac{16}{3}

Therefore, the coordinates of point $P$ are $(\frac{7}{2}, \frac{16}{3})$.

Conclusion

In this article, we explored how to find the coordinates of a point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$. We used the concept of ratios and proportions to solve for the coordinates of point $P$ and the distance from $R$ to $P$. This problem is a classic example of how to use ratios and proportions to solve problems in geometry and coordinate geometry.

Introduction

In our previous article, we explored how to find the coordinates of a point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$. In this article, we will answer some frequently asked questions related to this topic.

Q1: What is the formula for finding the coordinates of point $P$?

A1: The formula for finding the coordinates of point $P$ is:

x−x1x2−x1=56\frac{x - x_1}{x_2 - x_1} = \frac{5}{6}

y−y1y2−y1=56\frac{y - y_1}{y_2 - y_1} = \frac{5}{6}

Q2: How do I find the distance from $R$ to $P$?

A2: To find the distance from $R$ to $P$, you can use the formula:

d1=56d2d_1 = \frac{5}{6}d_2

where $d_2$ is the distance from $R$ to $Q$.

Q3: What if the coordinates of point $Q$ are not given?

A3: If the coordinates of point $Q$ are not given, you can use the concept of ratios and proportions to find the coordinates of point $P$. Let's assume that the coordinates of point $R$ are $(x_1, y_1)$ and the coordinates of point $P$ are $(x, y)$. We can then use the ratio $\frac{5}{6}$ to find the coordinates of point $P$.

Q4: Can I use this method to find the coordinates of point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{3}{4}$ the length of the line segment from $R$ to $Q$?

A4: Yes, you can use this method to find the coordinates of point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{3}{4}$ the length of the line segment from $R$ to $Q$. Simply replace the ratio $\frac{5}{6}$ with the ratio $\frac{3}{4}$ in the formulas.

Q5: What if the coordinates of point $R$ are not given?

A5: If the coordinates of point $R$ are not given, you can use the concept of ratios and proportions to find the coordinates of point $P$. Let's assume that the coordinates of point $P$ are $(x, y)$ and the coordinates of point $Q$ are $(x_2, y_2)$. We can then use the ratio $\frac{5}{6}$ to find the coordinates of point $P$.

Q6: Can I use this method to find the coordinates of point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{2}{3}$ the length of the line segment from $R$ to $Q$?

A6: Yes, you can use this method to find the coordinates of point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{2}{3}$ the length of the line segment from $R$ to $Q$. Simply replace the ratio $\frac{5}{6}$ with the ratio $\frac{2}{3}$ in the formulas.

Q7: What if the line segment from $R$ to $Q$ is not a straight line?

A7: If the line segment from $R$ to $Q$ is not a straight line, you can use the concept of ratios and proportions to find the coordinates of point $P$. However, you will need to use a different method to find the coordinates of point $P$, such as using the equation of a curve.

Q8: Can I use this method to find the coordinates of point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{1}{2}$ the length of the line segment from $R$ to $Q$?

A8: Yes, you can use this method to find the coordinates of point $P$ on a directed line segment from $R$ to $Q$, given that $P$ is $\frac{1}{2}$ the length of the line segment from $R$ to $Q$. Simply replace the ratio $\frac{5}{6}$ with the ratio $\frac{1}{2}$ in the formulas.

Conclusion

In this article, we answered some frequently asked questions related to finding the coordinates of point $P$ on a directed line segment from $R$ to $Q$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the concept.

Example Problems

Here are some example problems to help you practice finding the coordinates of point $P$ on a directed line segment from $R$ to $Q$:

  • Find the coordinates of point $P$ on a directed line segment from $R(1, 2)$ to $Q(4, 6)$, given that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$.
  • Find the coordinates of point $P$ on a directed line segment from $R(2, 3)$ to $Q(5, 7)$, given that $P$ is $\frac{3}{4}$ the length of the line segment from $R$ to $Q$.
  • Find the coordinates of point $P$ on a directed line segment from $R(3, 4)$ to $Q(6, 8)$, given that $P$ is $\frac{2}{3}$ the length of the line segment from $R$ to $Q$.

Solutions

Here are the solutions to the example problems:

  • Find the coordinates of point $P$ on a directed line segment from $R(1, 2)$ to $Q(4, 6)$, given that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$. The coordinates of point $P$ are $(\frac{7}{2}, \frac{16}{3})$.
  • Find the coordinates of point $P$ on a directed line segment from $R(2, 3)$ to $Q(5, 7)$, given that $P$ is $\frac{3}{4}$ the length of the line segment from $R$ to $Q$. The coordinates of point $P$ are $(\frac{11}{4}, \frac{23}{6})$.
  • Find the coordinates of point $P$ on a directed line segment from $R(3, 4)$ to $Q(6, 8)$, given that $P$ is $\frac{2}{3}$ the length of the line segment from $R$ to $Q$. The coordinates of point $P$ are $(\frac{7}{3}, \frac{16}{3})$.

Conclusion

In this article, we provided some example problems and solutions to help you practice finding the coordinates of point $P$ on a directed line segment from $R$ to $Q$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the concept.