Finding The X-coordinate Of A Point Using The Section Formula

In coordinate geometry, understanding how points divide line segments is crucial. One common concept is the section formula, which helps determine the coordinates of a point that divides a line segment in a given ratio. This article delves into a problem involving the section formula, providing a detailed explanation and solution. We will explore how to find the x-coordinate of a point given the coordinates of another point and the ratio in which a line segment is divided. Let's dive in and understand the intricacies of this problem.

Problem Statement

The problem we'll be addressing is as follows:

Point R divides the line segment ${\overline{PQ}}$ in the ratio 1:3. If the x-coordinate of R is -1 and the x-coordinate of P is -3, what is the x-coordinate of Q?

To solve this, we will use the section formula, a fundamental concept in coordinate geometry. This formula allows us to find the coordinates of a point that divides a line segment in a given ratio. Understanding and applying this formula correctly is essential for solving problems of this nature.

Understanding the Section Formula

The section formula is a vital tool in coordinate geometry that helps us determine the coordinates of a point which divides a line segment joining two points in a given ratio. Let's consider two points, ${P(x_1, y_1)}$ and ${Q(x_2, y_2)}$, and a point ${R}$ that divides the line segment ${\overline{PQ}}$ in the ratio ${m:n}$. The coordinates of point ${R}$, denoted as ${(x, y)}$, can be found using the following formulas:

${ x = \frac{mx_2 + nx_1}{m + n} }$

${ y = \frac{my_2 + ny_1}{m + n} }$

In our problem, we are primarily concerned with the x-coordinates, so we will focus on the first formula. The formula essentially calculates a weighted average of the x-coordinates of points ${P}$ and ${Q}$, where the weights are determined by the ratio ${m:n}$. The point ${R}$ is closer to ${P}$ if ${n}$ is greater than ${m}$, and vice versa. Understanding this formula is key to solving a variety of coordinate geometry problems, including the one we are tackling.

Applying the Section Formula to the Problem

In our specific problem, we are given that point R divides the line segment ${\overline{PQ}}$ in the ratio 1:3. This means that ${m = 1}$ and ${n = 3}$. We are also given that the x-coordinate of R is -1 and the x-coordinate of P is -3. Let's denote the x-coordinate of Q as ${x_Q}$. We can now substitute these values into the section formula for the x-coordinate:

${ -1 = \frac{1 \cdot x_Q + 3 \cdot (-3)}{1 + 3} }$

This equation represents the relationship between the x-coordinates of points P, Q, and R, based on the given ratio. To find the x-coordinate of Q, we need to solve this equation for ${x_Q}$. This involves algebraic manipulation to isolate ${x_Q}$ on one side of the equation. The next step is to simplify the equation and solve for ${x_Q}$.

Solving for the x-coordinate of Q

Now, let's proceed with solving the equation we derived from the section formula. The equation is:

${ -1 = \frac{1 \cdot x_Q + 3 \cdot (-3)}{1 + 3} }$

First, we simplify the denominator and the term in the numerator:

${ -1 = \frac{x_Q - 9}{4} }$

Next, we multiply both sides of the equation by 4 to eliminate the fraction:

${ -4 = x_Q - 9 }$

Now, we add 9 to both sides of the equation to isolate ${x_Q}$:

${ -4 + 9 = x_Q }$

${ 5 = x_Q }$

Therefore, the x-coordinate of point Q is 5. This result indicates that point Q lies to the right of point R on the x-axis, given the positions of points P and R and the division ratio. This step-by-step solution demonstrates the application of the section formula and the algebraic steps required to solve for the unknown coordinate.

Verification and Conclusion

To ensure our solution is correct, we can substitute the value we found for the x-coordinate of Q back into the section formula and check if it holds true. We found that the x-coordinate of Q is 5. Let's plug this value back into the formula:

${ x = \frac{mx_2 + nx_1}{m + n} }$

${ -1 = \frac{1 \cdot 5 + 3 \cdot (-3)}{1 + 3} }$

${ -1 = \frac{5 - 9}{4} }$

${ -1 = \frac{-4}{4} }$

${ -1 = -1 }$

The equation holds true, which confirms that our solution is correct. Therefore, the x-coordinate of point Q is indeed 5. In conclusion, by applying the section formula and performing the necessary algebraic steps, we have successfully determined the x-coordinate of point Q. This problem illustrates the importance of understanding and applying fundamental concepts in coordinate geometry. The correct answer is C. 5.

Key Takeaways

This problem serves as an excellent example of how the section formula can be applied to solve problems in coordinate geometry. The section formula is a fundamental tool for finding the coordinates of a point that divides a line segment in a given ratio. Understanding this formula and its applications is crucial for mastering coordinate geometry.

Importance of the Section Formula

The section formula is not just a mathematical equation; it's a powerful tool that helps us understand spatial relationships between points on a coordinate plane. It enables us to accurately determine the position of a point that divides a line segment in a specific ratio, which is essential in various fields such as engineering, computer graphics, and physics.

For instance, in computer graphics, the section formula can be used to interpolate between two points, creating smooth transitions and realistic animations. In physics, it can help determine the center of mass of a system of particles. The versatility of the section formula makes it a cornerstone of coordinate geometry and a valuable asset in problem-solving.

Step-by-Step Problem-Solving Approach

Solving the problem required a systematic, step-by-step approach. First, we identified the given information: the ratio in which point R divides the line segment ${\overline{PQ}}$, the x-coordinate of point R, and the x-coordinate of point P. Next, we applied the section formula, substituting the known values into the equation. Then, we simplified the equation and solved for the unknown x-coordinate of point Q. Finally, we verified our solution by substituting the value back into the original equation.

This methodical approach is crucial for tackling complex problems in mathematics and other disciplines. By breaking down a problem into smaller, manageable steps, we can avoid errors and arrive at the correct solution more efficiently. This problem-solving strategy can be applied to a wide range of mathematical problems, making it a valuable skill for students and professionals alike.

Common Mistakes to Avoid

When working with the section formula, there are several common mistakes that students often make. One of the most frequent errors is misinterpreting the ratio. It's essential to correctly identify which part of the ratio corresponds to which segment of the line. For example, in our problem, the ratio 1:3 means that the segment ${\overline{PR}}$ is one-third the length of the segment ${\overline{RQ}}$, not the entire line segment ${\overline{PQ}}$.

Another common mistake is incorrectly substituting values into the formula. It's crucial to ensure that the x-coordinates and the ratio values are placed in the correct positions in the equation. A simple way to avoid this is to write out the formula explicitly and then carefully substitute the values, double-checking each substitution.

Algebraic errors can also lead to incorrect solutions. When solving the equation, it's important to follow the correct order of operations and to be meticulous with each step. Multiplying both sides of the equation by the denominator, simplifying terms, and isolating the unknown variable should be done with care to avoid mistakes.

Finally, forgetting to verify the solution is a common oversight. Always substitute the value you find back into the original equation to check if it holds true. This step can catch errors and ensure that your answer is correct. By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving problems involving the section formula.

Practice Problems

To further solidify your understanding of the section formula, it's beneficial to practice with additional problems. Here are a few examples:

1. Point M divides the line segment ${\overline{AB}}$ in the ratio 2:5. If the x-coordinate of M is 3 and the x-coordinate of A is -2, what is the x-coordinate of B?
2. Point N divides the line segment ${\overline{CD}}$ in the ratio 3:4. If the x-coordinate of N is -1 and the x-coordinate of C is 2, what is the x-coordinate of D?
3. Point S divides the line segment ${\overline{PQ}}$ in the ratio 1:2. If the x-coordinate of S is 1 and the x-coordinate of P is -1, what is the x-coordinate of Q?

Working through these problems will help you become more comfortable with the section formula and its applications. Remember to follow the step-by-step problem-solving approach we discussed earlier and to verify your solutions. The more you practice, the more confident you will become in your ability to solve problems involving the section formula.

By mastering the section formula and its applications, you'll be well-equipped to tackle a wide range of problems in coordinate geometry and beyond. This fundamental concept is a building block for more advanced topics in mathematics, making it an essential skill for students and professionals alike.

Conclusion

In conclusion, understanding and applying the section formula is crucial for solving problems in coordinate geometry. This article has provided a detailed explanation of the section formula, a step-by-step solution to a specific problem, and key takeaways to reinforce your understanding. By mastering this concept, you'll be well-prepared to tackle a variety of mathematical challenges. Remember to practice regularly and apply the problem-solving strategies discussed to enhance your skills.