Using Section Formula to Find the Coordinates of Point C

Oct 13, 2025 by talynt.org 142 views

Introduction

In geometry, a line segment is a part of a line that is bounded by two distinct points. When a point divides a line segment in a given ratio, it is essential to determine the coordinates of that point. In this article, we will explore how to find the coordinates of a point that divides a line segment in the ratio 1:3.

Understanding the Problem

Consider the directed line segment ${$ AB $}$. We are given that point ${$ C $}$ divides the line segment in the ratio ${$ 1:3 $}$. This means that point ${$ C $}$ divides the line segment into two parts, one part being 1 unit and the other part being 3 units. Our goal is to find the coordinates of point ${$ C $}$.

Using Section Formula to Find the Coordinates of Point C

To find the coordinates of point ${$ C $}$, we can use the section formula. 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 this case, we have ${$ A(x_1, y_1) = A(2, 3) $}$ and ${$ B(x_2, y_2) = B(7, 1) $}$. We are given that point ${$ C $}$ divides the line segment in the ratio ${$ 1:3 $}$. Therefore, we can use the section formula to find the coordinates of point ${$ C $}$.

Applying the Section Formula

Substituting the values of ${$ A(x_1, y_1) = A(2, 3) $}$, ${$ B(x_2, y_2) = B(7, 1) $}$, and the ratio ${$ 1:3 $}$ into the section formula, we get:

${$ \left( \frac{1(7) + 3(2)}{1+3}, \frac{1(1) + 3(3)}{1+3} \right) $}$

Simplifying the expression, we get:

${$ \left( \frac{7 + 6}{4}, \frac{1 + 9}{4} \right) $}$

${$ \left( \frac{13}{4}, \frac{10}{4} \right) $}$

${$ \left( 3.25, 2.5 \right) $}$

However, this is not the only possible answer. We can also express the coordinates of point ${$ C $}$ as a fraction.

Expressing the Coordinates as a Fraction

We can express the coordinates of point ${$ C $}$ as a fraction by simplifying the expression:

${$ \left( \frac{13}{4}, \frac{10}{4} \right) $}$

${$ \left( \frac{13}{4}, \frac{5}{2} \right) $}$

Therefore, the coordinates of point ${$ C $}$ are ${$ \left( \frac{13}{4}, \frac{5}{2} \right) $}$.

Conclusion

In this article, we have explored how to find the coordinates of a point that divides a line segment in a given ratio. We used the section formula to find the coordinates of point ${$ C $}$ that divides the line segment ${$ AB $}$ in the ratio ${$ 1:3 $}$. We found that the coordinates of point ${$ C $}$ are ${$ \left( \frac{13}{4}, \frac{5}{2} \right) $}$. This demonstrates the importance of using the section formula to find the coordinates of a point that divides a line segment in a given ratio.

Final Answer

The final answer is ${$ \left( \frac{13}{4}, \frac{5}{2} \right) $}$.

Introduction

In our previous article, we explored how to find the coordinates of a point that divides a line segment in a given ratio. We used the section formula to find the coordinates of point ${$ C $}$ that divides the line segment ${$ AB $}$ in the ratio ${$ 1:3 $}$. In this article, we will answer some frequently asked questions related to finding the coordinates of a point that divides a line segment in a given ratio.

Q1: What is the section formula?

A1: The section formula is a formula used to find the coordinates of a point that divides a line segment in a given ratio. It 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) $}$

Q2: How do I apply the section formula to find the coordinates of a point that divides a line segment in a given ratio?

A2: To apply the section formula, you need to substitute the values of ${$ A(x_1, y_1) $}$, ${$ B(x_2, y_2) $}$, and the ratio ${$ m:n $}$ into the formula. Then, simplify the expression to find the coordinates of the point.

Q3: What if the ratio is not in the simplest form?

A3: If the ratio is not in the simplest form, you can simplify it before applying the section formula. For example, if the ratio is ${$ 2:6 $}$, you can simplify it to ${$ 1:3 $}$.

Q4: Can I use the section formula to find the coordinates of a point that divides a line segment in a given ratio if the line segment is not parallel to the x-axis or y-axis?

A4: Yes, you can use the section formula to find the coordinates of a point that divides a line segment in a given ratio even if the line segment is not parallel to the x-axis or y-axis. The section formula works for any line segment, regardless of its orientation.

Q5: How do I determine the coordinates of a point that divides a line segment in a given ratio if the line segment is not in the standard form (x, y)?

A5: If the line segment is not in the standard form (x, y), you can convert it to the standard form by finding the equation of the line that passes through the two points. Then, you can use the section formula to find the coordinates of the point that divides the line segment in the given ratio.

Q6: Can I use the section formula to find the coordinates of a point that divides a line segment in a given ratio if the line segment is a circle?

A6: No, you cannot use the section formula to find the coordinates of a point that divides a line segment in a given ratio if the line segment is a circle. The section formula is only applicable to line segments, not circles.

Q7: How do I find the coordinates of a point that divides a line segment in a given ratio if the line segment is a line with a slope?

A7: To find the coordinates of a point that divides a line segment in a given ratio if the line segment is a line with a slope, you can use the point-slope form of a line. Then, you can use the section formula to find the coordinates of the point that divides the line segment in the given ratio.

Conclusion

In this article, we have answered some frequently asked questions related to finding the coordinates of a point that divides a line segment in a given ratio. We have covered topics such as the section formula, applying the section formula, simplifying ratios, and using the section formula for line segments that are not parallel to the x-axis or y-axis. We hope that this article has been helpful in clarifying any doubts you may have had about finding the coordinates of a point that divides a line segment in a given ratio.

Final Answer

The final answer is that the section formula is a powerful tool for finding the coordinates of a point that divides a line segment in a given ratio. By understanding how to apply the section formula and simplifying ratios, you can easily find the coordinates of a point that divides a line segment in a given ratio.