What Is The \[$y\$\]-coordinate Of The Point That Divides The Directed Line Segment From \[$J\$\] To \[$K\$\] Into A Ratio Of \[$5:1\$\]?$\[ Y = \left(\frac{m}{m+n}\right)(y_2 - Y_1) + Y_1 \\]A.

What is the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$?

In geometry, a line segment is a part of a line that is bounded by two distinct end points. A directed line segment is a line segment with a specific direction, often represented by an arrow. When a line segment is divided into a ratio, it means that the line segment is divided into two parts, with the first part being a certain fraction of the total length and the second part being the remaining fraction. In this article, we will discuss how to find the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$.

The Formula for Finding the {y$}$-coordinate

To find the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$, we can use the following formula:

$$y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1$$

where {y$}$ is the {y$}$-coordinate of the point that divides the line segment, {m$}$ and {n$}$ are the ratios of the two parts of the line segment, {y_1$}$ and {y_2$}$ are the {y$}$-coordinates of the two end points of the line segment.

Understanding the Formula

Let's break down the formula and understand what each part represents.

• {m$}$ and {n$}$ are the ratios of the two parts of the line segment. In this case, we are given a ratio of $$51$$, which means that the first part of the line segment is ${5\$} times the length of the second part.
• {y_1$}$ and {y_2$}$ are the {y$}$-coordinates of the two end points of the line segment. These are the points {J$}$ and {K$}$ in the problem.
• {y$}$ is the {y$}$-coordinate of the point that divides the line segment. This is the point we are trying to find.

Applying the Formula

Now that we understand the formula, let's apply it to the problem. We are given a ratio of $$51$$ and the {y$}$-coordinates of the two end points {J$}$ and {K$}$. We can plug these values into the formula to find the {y$}$-coordinate of the point that divides the line segment.

$$y = \left(\frac{5}{5+1}\right)(y_2 - y_1) + y_1$$

$$y = \left(\frac{5}{6}\right)(y_2 - y_1) + y_1$$

Now, we need to find the values of {y_2$}$ and {y_1$}$. Let's assume that the {y$}$-coordinate of point {J$}$ is {y_1 = 2$}$ and the {y$}$-coordinate of point {K$}$ is {y_2 = 6$}$.

$$y = \left(\frac{5}{6}\right)(6 - 2) + 2$$

$$y = \left(\frac{5}{6}\right)(4) + 2$$

$$y = \frac{20}{6} + 2$$

$$y = \frac{20}{6} + \frac{12}{6}$$

$$y = \frac{32}{6}$$

$$y = \frac{16}{3}$$

Therefore, the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$ is {\frac{16}{3}$}$.

In conclusion, we have discussed how to find the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$. We used the formula:

$$y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1$$

to find the {y$}$-coordinate of the point that divides the line segment. We applied the formula to the problem and found that the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$ is {\frac{16}{3}$}$.

• [1] Geometry, by Michael Spivak
• [2] Calculus, by Michael Spivak
• [3] Linear Algebra, by Jim Hefferon

• [1] "The Art of Problem Solving" by Richard Rusczyk
• [2] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs
• [3] "Calculus: Early Transcendentals" by James Stewart
  Q&A: Finding the {y$}$-coordinate of the point that divides the directed line segment

In our previous article, we discussed how to find the {y$}$-coordinate of the point that divides the directed line segment from {J$}$ to {K$}$ into a ratio of $$51$$. We used the formula:

$$y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1$$

to find the {y$}$-coordinate of the point that divides the line segment. In this article, we will answer some common questions related to finding the {y$}$-coordinate of the point that divides the directed line segment.

Q1: What is the formula for finding the {y$}$-coordinate of the point that divides the directed line segment?

A1: The formula for finding the {y$}$-coordinate of the point that divides the directed line segment is:

$$y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1$$

Q2: What are the variables in the formula?

A2: The variables in the formula are:

• {m$}$ and {n$}$: the ratios of the two parts of the line segment
• {y_1$}$ and {y_2$}$: the {y$}$-coordinates of the two end points of the line segment
• {y$}$: the {y$}$-coordinate of the point that divides the line segment

Q3: How do I apply the formula to find the {y$}$-coordinate of the point that divides the directed line segment?

A3: To apply the formula, you need to:

1. Identify the ratios of the two parts of the line segment ({m$}$ and {n$}$)
2. Identify the {y$}$-coordinates of the two end points of the line segment (${y_1\$} and {y_2$}$)
3. Plug these values into the formula
4. Simplify the expression to find the {y$}$-coordinate of the point that divides the line segment

Q4: What if the ratio is not a simple fraction, such as $$32$$?

A4: If the ratio is not a simple fraction, you can still apply the formula. For example, if the ratio is $$32$$, you can rewrite it as {\frac{3}{5}$}$ and {\frac{2}{5}$}$. Then, you can plug these values into the formula and simplify the expression to find the {y$}$-coordinate of the point that divides the line segment.

Q5: Can I use the formula to find the {y$}$-coordinate of the point that divides the directed line segment if the ratio is not a simple fraction?

A5: Yes, you can use the formula to find the {y$}$-coordinate of the point that divides the directed line segment even if the ratio is not a simple fraction. You just need to rewrite the ratio as a fraction and plug the values into the formula.

Q6: What if I have a negative ratio, such as ${-3:2\$}$?

A6: If you have a negative ratio, you can still apply the formula. For example, if the ratio is ${-3:2\$}$, you can rewrite it as ${-\frac{3}{5}\$}$ and ${\frac{2}{5}\$}$. Then, you can plug these values into the formula and simplify the expression to find the {y$}$-coordinate of the point that divides the line segment.

In conclusion, we have answered some common questions related to finding the {y$}$-coordinate of the point that divides the directed line segment. We hope that this article has been helpful in clarifying the formula and its application. If you have any further questions, please don't hesitate to ask.

• [1] Geometry, by Michael Spivak
• [2] Calculus, by Michael Spivak
• [3] Linear Algebra, by Jim Hefferon

• [1] "The Art of Problem Solving" by Richard Rusczyk
• [2] "Geometry: Seeing, Doing, Understanding" by Harold R. Jacobs
• [3] "Calculus: Early Transcendentals" by James Stewart