Applying the Section Formula to the Given Problem

Oct 13, 2025 by windailys.com 42 views

Introduction

When dealing with directed line segments on a number line, it's essential to understand how to partition them in a given ratio. In this scenario, we're given the endpoints of the directed line segment from Q to S as -14 and 2, respectively. Point R partitions this directed line segment in a 3:5 ratio. Our goal is to find the correct expression that uses the formula to determine the position of point R.

Understanding the Ratio

To begin, let's understand the concept of a ratio. A ratio is a way of comparing two numbers by division. In this case, the ratio of the directed line segment from Q to S is 3:5. This means that for every 3 units of distance from Q, there are 5 units of distance from R to S.

Using the Section Formula

The section formula is a mathematical concept used to find the coordinates of a point that divides a line segment in a given ratio. The formula is as follows:

If a point P(x, y) divides the line segment joining the points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of P are given by:

x = (mx2 + nx1)/(m+n) y = (my2 + ny1)/(m+n)

Applying the Section Formula to the Given Problem

In this scenario, we're given the endpoints of the directed line segment from Q to S as -14 and 2, respectively. Point R partitions this directed line segment in a 3:5 ratio. Using the section formula, we can find the position of point R.

Let's assume the coordinates of point R are (x, y). Since point R divides the directed line segment from Q to S in a 3:5 ratio, we can use the section formula to find the coordinates of point R.

x = (3(2) + 5(-14))/(3+5) x = (6 - 70)/8 x = -64/8 x = -8

y = (3(2) + 5(-14))/(3+5) y = (6 - 70)/8 y = -64/8 y = -8

Finding the Correct Expression

Now that we have found the coordinates of point R, we can express it as an ordered pair (x, y). However, we need to find the correct expression that uses the formula to determine the position of point R.

The correct expression is:

(-8, -8)

This expression correctly uses the formula to find the position of point R, which partitions the directed line segment from Q to S in a 3:5 ratio.

Conclusion

In conclusion, we have successfully used the section formula to find the position of point R, which partitions the directed line segment from Q to S in a 3:5 ratio. The correct expression that uses the formula to determine the position of point R is (-8, -8).

Frequently Asked Questions

What is the ratio of the directed line segment from Q to S? The ratio of the directed line segment from Q to S is 3:5.
What is the position of point R? The position of point R is (-8, -8).
What is the formula used to find the position of point R? The formula used to find the position of point R is the section formula.

Final Answer

The final answer is $\boxed{(-8, -8)}$ .

Introduction

In our previous article, we discussed how to find the position of point R, which partitions the directed line segment from Q to S in a 3:5 ratio. We used the section formula to determine the coordinates of point R. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the ratio of the directed line segment from Q to S?

A1: The ratio of the directed line segment from Q to S is 3:5.

Q2: What is the position of point R?

A2: The position of point R is (-8, -8).

Q3: What is the formula used to find the position of point R?

A3: The formula used to find the position of point R is the section formula.

Q4: How do I apply the section formula to find the position of point R?

A4: To apply the section formula, you need to use the following formula:

x = (mx2 + nx1)/(m+n) y = (my2 + ny1)/(m+n)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment, and m:n is the ratio in which the line segment is divided.

Q5: What if the ratio is not in the simplest form?

A5: If the ratio is not in the simplest form, you can simplify it by dividing both numbers by their greatest common divisor (GCD).

Q6: Can I use the section formula to find the position of point R if the ratio is not 3:5?

A6: Yes, you can use the section formula to find the position of point R for any ratio.

Q7: How do I find the coordinates of point R if the ratio is 2:3?

A7: To find the coordinates of point R if the ratio is 2:3, you can use the section formula as follows:

x = (2(2) + 3(-14))/(2+3) x = (4 - 42)/5 x = -38/5 x = -7.6

y = (2(2) + 3(-14))/(2+3) y = (4 - 42)/5 y = -38/5 y = -7.6

Q8: Can I use the section formula to find the position of point R if the line segment is not horizontal?

A8: Yes, you can use the section formula to find the position of point R if the line segment is not horizontal.

Q9: How do I find the coordinates of point R if the line segment is not horizontal?

A9: To find the coordinates of point R if the line segment is not horizontal, you need to use the section formula as follows:

x = (mx2 + nx1)/(m+n) y = (my2 + ny1)/(m+n)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment, and m:n is the ratio in which the line segment is divided.

Q10: Can I use the section formula to find the position of point R if the line segment is vertical?

A10: Yes, you can use the section formula to find the position of point R if the line segment is vertical.

Conclusion

In conclusion, we have answered some frequently asked questions related to the topic of finding the position of point R, which partitions the directed line segment from Q to S in a 3:5 ratio. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is $\boxed{(-8, -8)}$ .