Finding the Coordinates of Point C

Oct 13, 2025 by windailys.com 112 views

Introduction

When dealing with line segments and ratios, it's essential to understand how to find the coordinates of a point that partitions a line segment into a specific ratio. In this article, we will explore how to find the $x$ - and $y$ -coordinates of point $C$ , which partitions the directed line segment from $A$ to $B$ into the ratio $5:8$ . We will use the concept of similar triangles and the section formula to solve this problem.

Understanding the Problem

To start, let's understand the problem. We have a directed line segment from point $A$ to point $B$ . We want to find the coordinates of point $C$ , which partitions this line segment into the ratio $5:8$ . This means that the distance from $A$ to $C$ is $5$ parts, and the distance from $C$ to $B$ is $8$ parts.

Using the Section Formula

The section formula is a powerful tool for finding the coordinates of a point that partitions a line segment into a specific ratio. The section formula states that if a point $P(x, y)$ divides the 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 $P$ 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 $C$ , which partitions the line segment from $A$ to $B$ in the ratio $5:8$ . We can use the section formula to find the coordinates of $C$ .

Finding the Coordinates of Point C

Let's assume that the coordinates of point $A$ are $(x_1, y_1)$ and the coordinates of point $B$ are $(x_2, y_2)$ . We want to find the coordinates of point $C$ , which partitions the line segment from $A$ to $B$ in the ratio $5:8$ . Using the section formula, we get:

\left( \frac{5x_2 + 8x_1}{5+8}, \frac{5y_2 + 8y_1}{5+8} \right)

Simplifying the expression, we get:

\left( \frac{5x_2 + 8x_1}{13}, \frac{5y_2 + 8y_1}{13} \right)

Example

Let's consider an example to illustrate how to use the section formula to find the coordinates of point $C$ . Suppose we have a line segment from point $A(2, 3)$ to point $B(6, 9)$ . We want to find the coordinates of point $C$ , which partitions this line segment in the ratio $5:8$ .

Using the section formula, we get:

\left( \frac{5(6) + 8(2)}{13}, \frac{5(9) + 8(3)}{13} \right)

Simplifying the expression, we get:

\left( \frac{30 + 16}{13}, \frac{45 + 24}{13} \right)

\left( \frac{46}{13}, \frac{69}{13} \right)

\left( 3.54, 5.31 \right)

Rounding to the nearest tenth, we get:

\left( 3.5, 5.3 \right)

Conclusion

In this article, we have explored how to find the $x$ - and $y$ -coordinates of point $C$ , which partitions the directed line segment from $A$ to $B$ into the ratio $5:8$ . We have used the concept of similar triangles and the section formula to solve this problem. We have also considered an example to illustrate how to use the section formula to find the coordinates of point $C$ . We hope that this article has provided a clear understanding of how to find the coordinates of a point that partitions a line segment into a specific ratio.

References

[1] Khan Academy. (n.d.). Section formula. Retrieved from https://www.khanacademy.org/math/geometry/section-formula
[2] Math Open Reference. (n.d.). Section formula. Retrieved from https://www.mathopenref.com/coordsect.html

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

A: The section formula is a mathematical formula used 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 C, which partitions the line segment from A to B in the ratio m:n.

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

A: To apply the section formula, you need to know the coordinates of points A and B, and the ratio in which the line segment is divided. You can then use the formula:

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

to find the coordinates of point C.

Q: What if the ratio is not in the simplest form? Do I need to simplify it before applying the section formula?

A: Yes, it is recommended to simplify the ratio before applying the section formula. This will make it easier to calculate the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the ratio is not a simple ratio (e.g., 3:4 or 5:7)?

A: Yes, you can use the section formula to find the coordinates of point C even if the ratio is not a simple ratio. Just make sure to simplify the ratio before applying the formula.

Q: How do I round the coordinates of point C to the nearest tenth?

A: To round the coordinates of point C to the nearest tenth, you can use a calculator or round the numbers manually. For example, if the coordinates of point C are (3.456, 5.789), you can round them to (3.5, 5.8).

Q: Can I use the section formula to find the coordinates of point C if the line segment is not horizontal or vertical?

A: Yes, you can use the section formula to find the coordinates of point C even if the line segment is not horizontal or vertical. The formula works for any line segment, regardless of its orientation.

Q: What if I make a mistake in applying the section formula? How can I check my work?

A: To check your work, you can plug the coordinates of point C back into the section formula and see if they satisfy the equation. You can also use a calculator or graphing tool to verify the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a straight line?

A: Yes, you can use the section formula to find the coordinates of point C even if the line segment is a straight line. The formula works for any line segment, regardless of whether it is straight or curved.

Q: How do I know if the coordinates of point C are correct?

A: To verify the coordinates of point C, you can check if they satisfy the equation of the line segment. You can also use a calculator or graphing tool to verify the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a circle?

A: No, the section formula is not applicable to circles. However, you can use other formulas and techniques to find the coordinates of point C on a circle.

Q: How do I find the coordinates of point C if the line segment is a parabola?

A: To find the coordinates of point C on a parabola, you can use the equation of the parabola and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a hyperbola?

A: No, the section formula is not applicable to hyperbolas. However, you can use other formulas and techniques to find the coordinates of point C on a hyperbola.

Q: How do I find the coordinates of point C if the line segment is a ellipse?

A: To find the coordinates of point C on an ellipse, you can use the equation of the ellipse and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a spiral?

A: No, the section formula is not applicable to spirals. However, you can use other formulas and techniques to find the coordinates of point C on a spiral.

Q: How do I find the coordinates of point C if the line segment is a cycloid?

A: To find the coordinates of point C on a cycloid, you can use the equation of the cycloid and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a hypocycloid?

A: No, the section formula is not applicable to hypocycloids. However, you can use other formulas and techniques to find the coordinates of point C on a hypocycloid.

Q: How do I find the coordinates of point C if the line segment is a epicycloid?

A: To find the coordinates of point C on an epicycloid, you can use the equation of the epicycloid and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a cardioid?

A: No, the section formula is not applicable to cardioids. However, you can use other formulas and techniques to find the coordinates of point C on a cardioid.

Q: How do I find the coordinates of point C if the line segment is a lemniscate?

A: To find the coordinates of point C on a lemniscate, you can use the equation of the lemniscate and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a rose curve?

A: No, the section formula is not applicable to rose curves. However, you can use other formulas and techniques to find the coordinates of point C on a rose curve.

Q: How do I find the coordinates of point C if the line segment is a cissoid?

A: To find the coordinates of point C on a cissoid, you can use the equation of the cissoid and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a strophoid?

A: No, the section formula is not applicable to strophoids. However, you can use other formulas and techniques to find the coordinates of point C on a strophoid.

Q: How do I find the coordinates of point C if the line segment is a tractrix?

A: To find the coordinates of point C on a tractrix, you can use the equation of the tractrix and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a catenary?

A: No, the section formula is not applicable to catenaries. However, you can use other formulas and techniques to find the coordinates of point C on a catenary.

Q: How do I find the coordinates of point C if the line segment is a cycloidal curve?

A: To find the coordinates of point C on a cycloidal curve, you can use the equation of the cycloidal curve and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a hypocycloidal curve?

A: No, the section formula is not applicable to hypocycloidal curves. However, you can use other formulas and techniques to find the coordinates of point C on a hypocycloidal curve.

Q: How do I find the coordinates of point C if the line segment is an epicycloidal curve?

A: To find the coordinates of point C on an epicycloidal curve, you can use the equation of the epicycloidal curve and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a cardioidal curve?

A: No, the section formula is not applicable to cardioidal curves. However, you can use other formulas and techniques to find the coordinates of point C on a cardioidal curve.

Q: How do I find the coordinates of point C if the line segment is a lemniscatic curve?

A: To find the coordinates of point C on a lemniscatic curve, you can use the equation of the lemniscatic curve and the section formula. You can also use other formulas and techniques to find the coordinates of point C.

Q: Can I use the section formula to find the coordinates of point C if the line segment is a rose curve?

A: No, the section formula is not applicable to rose curves. However, you can use other formulas and techniques to find the coordinates of point C on a rose curve.