Incorrectly substituting the values of the coordinates of points A and B

Oct 13, 2025 by windailys.com 36 views

Introduction

When dealing with directed line segments, it's often necessary to find the coordinates of a point that divides the line segment into a specific ratio. In this case, we're looking for the $x$ - and $y$ -coordinates of point $P$ on the directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ the length of the line segment. This problem can be solved using the concept of similar triangles and the section formula.

Understanding the Problem

To start, let's understand the problem visually. We have a directed line segment from point $A$ to point $B$ . We want to find the coordinates of point $P$ such that it divides the line segment into a ratio of $\frac{2}{3}$ . This means that point $P$ will be $\frac{2}{3}$ of the distance from point $A$ to point $B$ .

Using the Section Formula

The section formula is a mathematical concept that allows us to find the coordinates of a point that divides a line segment into a specific ratio. The section formula states that if a point $P$ divides a line segment joining the points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$ , then the coordinates of point $P$ are given by:

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

In our case, we want to find the coordinates of point $P$ such that it divides the line segment from $A$ to $B$ in the ratio $\frac{2}{3}$ . Using the section formula, we can write:

\left(\frac{2x_2 + 3x_1}{2+3}, \frac{2y_2 + 3y_1}{2+3}\right)

Simplifying the Expression

Now, let's simplify the expression for the coordinates of point $P$ . We can start by evaluating the denominator:

2+3 = 5

So, the expression for the coordinates of point $P$ becomes:

\left(\frac{2x_2 + 3x_1}{5}, \frac{2y_2 + 3y_1}{5}\right)

Finding the Coordinates of Point $P$

Now that we have the expression for the coordinates of point $P$ , we can find the actual coordinates by substituting the values of $x_1$ , $y_1$ , $x_2$ , and $y_2$ . Let's assume that the coordinates of point $A$ are $(x_1, y_1)$ and the coordinates of point $B$ are $(x_2, y_2)$ .

Substituting these values into the expression for the coordinates of point $P$ , we get:

\left(\frac{2x_2 + 3x_1}{5}, \frac{2y_2 + 3y_1}{5}\right)

Conclusion

In this article, we have shown how to find the $x$ - and $y$ -coordinates of point $P$ on the directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ the length of the line segment. We used the concept of similar triangles and the section formula to derive the expression for the coordinates of point $P$ . By simplifying the expression and substituting the values of the coordinates of points $A$ and $B$ , we can find the actual coordinates of point $P$ .

Example

Let's consider an example to illustrate the concept. Suppose we have a directed line segment from point $A(2, 3)$ to point $B(6, 9)$ . We want to find the coordinates of point $P$ such that it divides the line segment in the ratio $\frac{2}{3}$ .

Using the section formula, we can find the coordinates of point $P$ as:

\left(\frac{2(6) + 3(2)}{5}, \frac{2(9) + 3(3)}{5}\right)

Simplifying the expression, we get:

\left(\frac{12 + 6}{5}, \frac{18 + 9}{5}\right)

\left(\frac{18}{5}, \frac{27}{5}\right)

Therefore, the coordinates of point $P$ are $(\frac{18}{5}, \frac{27}{5})$ .

Applications

The concept of finding the coordinates of a point that divides a line segment into a specific ratio has numerous applications in mathematics and real-world scenarios. Some of the applications include:

Geometry: The concept is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in geometry.
Trigonometry: The concept is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in trigonometry.
Physics: The concept is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in physics.
Engineering: The concept is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in engineering.

Final Thoughts

In conclusion, finding the $x$ - and $y$ -coordinates of point $P$ on the directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ the length of the line segment is a fundamental concept in mathematics. The concept is used to find the coordinates of points that divide line segments in a specific ratio, which has numerous applications in mathematics and real-world scenarios. By understanding the concept and using the section formula, we can find the coordinates of point $P$ and solve various problems in mathematics and real-world scenarios.

Q: What is the section formula, and how is it used to find the coordinates of point P?

A: The section formula is a mathematical concept that allows us to find the coordinates of a point that divides a line segment into a specific ratio. It is used to find the coordinates of point P by substituting the values of the coordinates of points A and B into the formula.

Q: How do I apply the section formula to find the coordinates of point P?

A: To apply the section formula, you need to substitute the values of the coordinates of points A and B into the formula. The formula is given by:

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

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

Q: What is the significance of the ratio m:n in the section formula?

A: The ratio m:n in the section formula represents the ratio in which the line segment is divided. For example, if the ratio is 2:3, it means that point P divides the line segment into two parts, one part being 2/5 of the total length and the other part being 3/5 of the total length.

Q: Can I use the section formula to find the coordinates of point P if the ratio is not a simple fraction?

A: Yes, you can use the section formula to find the coordinates of point P even if the ratio is not a simple fraction. The formula will still work as long as you substitute the correct values of the coordinates of points A and B and the ratio m:n.

Q: How do I simplify the expression for the coordinates of point P?

A: To simplify the expression for the coordinates of point P, you need to evaluate the denominator and simplify the numerator. For example, if the denominator is 5, you can simplify the expression by dividing both the numerator and the denominator by 5.

Q: Can I use the section formula to find the coordinates of point P if the line segment is not directed?

A: Yes, you can use the section formula to find the coordinates of point P even if the line segment is not directed. The formula will still work as long as you substitute the correct values of the coordinates of points A and B and the ratio m:n.

Q: What are some real-world applications of the section formula?

A: The section formula has numerous real-world applications, including:

Geometry: The formula is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in geometry.
Trigonometry: The formula is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in trigonometry.
Physics: The formula is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in physics.
Engineering: The formula is used to find the coordinates of points that divide line segments in a specific ratio, which is essential in engineering.

Q: How do I use the section formula to find the coordinates of point P in a 3D space?

A: To use the section formula to find the coordinates of point P in a 3D space, you need to substitute the values of the coordinates of points A and B into the formula. The formula is given by:

\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)

where (x1, y1, z1) are the coordinates of point A, (x2, y2, z2) are the coordinates of point B, and m:n is the ratio in which the line segment is divided.

Q: Can I use the section formula to find the coordinates of point P if the line segment is not a straight line?

A: No, you cannot use the section formula to find the coordinates of point P if the line segment is not a straight line. The formula is only applicable to straight line segments.

Q: How do I find the coordinates of point P if the ratio is not a simple fraction?

A: To find the coordinates of point P if the ratio is not a simple fraction, you need to substitute the values of the coordinates of points A and B into the formula and simplify the expression. The formula is given by:

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

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

Q: Can I use the section formula to find the coordinates of point P if the line segment is not a directed line segment?

A: Yes, you can use the section formula to find the coordinates of point P even if the line segment is not a directed line segment. The formula will still work as long as you substitute the correct values of the coordinates of points A and B and the ratio m:n.

Q: What are some common mistakes to avoid when using the section formula?

A: Some common mistakes to avoid when using the section formula include:

Incorrectly substituting the values of the coordinates of points A and B: Make sure to substitute the correct values of the coordinates of points A and B into the formula.
Incorrectly simplifying the expression: Make sure to simplify the expression correctly by evaluating the denominator and simplifying the numerator.
Not using the correct ratio: Make sure to use the correct ratio m:n in the formula.

Q: How do I verify the correctness of the coordinates of point P?

A: To verify the correctness of the coordinates of point P, you need to check that the coordinates satisfy the given ratio and that the line segment is divided into the correct ratio. You can also use the distance formula to verify that the coordinates of point P are correct.