Finding The Y-coordinate Dividing A Line Segment In A 2:3 Ratio

$$$$

In the realm of coordinate geometry, the section formula stands as a pivotal concept, enabling us to pinpoint the coordinates of a point that divides a line segment in a given ratio. This formula finds extensive applications in various mathematical domains, including geometry, calculus, and linear algebra. In this comprehensive guide, we will delve into the intricacies of the section formula, elucidating its derivation, applications, and practical usage through a step-by-step approach. Our focus will be on determining the $y$-coordinate of a point that divides a directed line segment from point $J$ to point $K$ in a specific ratio of 2:3. Let's embark on this journey of mathematical exploration and unravel the secrets of the section formula.

Delving into the Section Formula

The section formula is a cornerstone of coordinate geometry, providing a precise method for calculating the coordinates of a point that partitions a line segment in a specific ratio. This formula holds immense significance in numerous mathematical contexts, including geometry, calculus, and linear algebra. To fully grasp the concept, let's consider two points, $J(x_1, y_1)$ and $K(x_2, y_2)$, which define a line segment. Our objective is to determine the coordinates of a point, say $P(x, y)$, that divides this line segment internally in the ratio $m:n$. This means that the distance from $J$ to $P$ is $m$ times some constant, and the distance from $P$ to $K$ is $n$ times the same constant. Mathematically, we can express this relationship as:

$\frac{JP}{PK} = \frac{m}{n}$

The section formula provides the coordinates of point $P$ as follows:

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

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

These formulas elegantly express the coordinates of the point $P$ in terms of the coordinates of the endpoints $J$ and $K$, and the ratio $m:n$. The $x$-coordinate of $P$ is a weighted average of the $x$-coordinates of $J$ and $K$, with weights $n$ and $m$, respectively, divided by the sum of the weights. Similarly, the $y$-coordinate of $P$ is a weighted average of the $y$-coordinates of $J$ and $K$, with the same weights. This weighted average interpretation is crucial for understanding the formula's behavior and its connection to the concept of dividing a line segment proportionally. When $m = n$, the point $P$ becomes the midpoint of the line segment $JK$, and the section formula simplifies to the midpoint formula, which we will discuss later.

Derivation of the Section Formula

To fully appreciate the power and elegance of the section formula, it's essential to understand its derivation. The derivation hinges on the principles of similar triangles and the properties of proportions. Let's embark on a step-by-step journey to unravel the derivation of this fundamental formula.

Consider points $J(x_1, y_1)$ and $K(x_2, y_2)$ in a coordinate plane, and let $P(x, y)$ be the point that divides the line segment $JK$ internally in the ratio $m:n$. This means that the ratio of the lengths of the segments $JP$ and $PK$ is equal to $m/n$.

To begin the derivation, we construct perpendiculars from points $J$, $P$, and $K$ to the x-axis, labeling the feet of these perpendiculars as $A$, $B$, and $C$, respectively. Similarly, we draw perpendiculars from $J$ and $P$ to the line segment $KC$, labeling the feet of these perpendiculars as $D$ and $E$, respectively.

Now, let's analyze the triangles formed in this construction. We can identify two key triangles: $\triangle JDP$ and $\triangle PEK$. These triangles are similar because their corresponding angles are equal. Specifically, $\angle JPD = \angle KPE$ (vertically opposite angles), $\angle JDP = \angle PEK$ (both are right angles), and $\angle DJP = \angle EKP$ (corresponding angles). The similarity of these triangles is the cornerstone of the derivation.

Since $\triangle JDP$ and $\triangle PEK$ are similar, the ratios of their corresponding sides are equal. This leads to the following proportions:

$\frac{JD}{PE} = \frac{JP}{PK} = \frac{DP}{EK} = \frac{m}{n}$

Now, let's express the lengths of the line segments in terms of the coordinates of the points. We have:

• $JD = AB = x - x_1$
• $PE = BC = x_2 - x$
• $DP = PB = y - y_1$
• $EK = KC = y_2 - y$

Substituting these expressions into the proportions, we get:

$\frac{x - x_1}{x_2 - x} = \frac{m}{n}$ and $\frac{y - y_1}{y_2 - y} = \frac{m}{n}$

These equations relate the coordinates of point $P$ to the coordinates of points $J$ and $K$, and the ratio $m:n$. To find the coordinates of $P$, we solve these equations for $x$ and $y$.

Cross-multiplying the first equation, we get:

$n(x - x_1) = m(x_2 - x)$

Expanding and rearranging the terms, we have:

$nx - nx_1 = mx_2 - mx$

$(m + n)x = mx_2 + nx_1$

Dividing both sides by $(m + n)$, we obtain the x-coordinate of $P$:

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

Similarly, cross-multiplying the second equation, we get:

$n(y - y_1) = m(y_2 - y)$

Expanding and rearranging the terms, we have:

$ny - ny_1 = my_2 - my$

$(m + n)y = my_2 + ny_1$

Dividing both sides by $(m + n)$, we obtain the y-coordinate of $P$:

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

Thus, we have derived the section formula, which provides the coordinates of the point $P(x, y)$ that divides the line segment $JK$ in the ratio $m:n$. The derivation relies on the fundamental principles of similar triangles and proportions, showcasing the interconnectedness of geometric concepts.

Applying the Section Formula: Finding the $y$-Coordinate in a 2:3 Ratio

Now that we have a solid understanding of the section formula and its derivation, let's apply it to solve the specific problem at hand: determining the $y$-coordinate of the point that divides the directed line segment from $J$ to $K$ in the ratio of 2:3. To do this, we'll assume that we have the coordinates of points $J$ and $K$. Let $J$ have coordinates $(x_1, y_1)$ and $K$ have coordinates $(x_2, y_2)$. The ratio $m:n$ is given as 2:3, which means $m = 2$ and $n = 3$.

We are interested in finding the $y$-coordinate of the point that divides the segment $JK$ in this ratio. Using the section formula for the $y$-coordinate, we have:

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

Substitute the given values of $m$ and $n$ into the formula:

$y = \frac{2y_2 + 3y_1}{2 + 3}$

Simplify the expression:

$y = \frac{2y_2 + 3y_1}{5}$

This formula gives us the $y$-coordinate of the point that divides the line segment $JK$ in the ratio 2:3. To obtain a numerical value for $y$, we need the specific coordinates of points $J$ and $K$. Let's consider a few examples to illustrate the application of this formula.

Example 1:

Suppose $J$ has coordinates (1, 2) and $K$ has coordinates (6, 7). Then, $x_1 = 1$, $y_1 = 2$, $x_2 = 6$, and $y_2 = 7$. Plugging these values into the formula for the $y$-coordinate, we get:

$y = \frac{2(7) + 3(2)}{5} = \frac{14 + 6}{5} = \frac{20}{5} = 4$

Therefore, the $y$-coordinate of the point that divides the line segment $JK$ in the ratio 2:3 is 4.

Example 2:

Suppose $J$ has coordinates (-2, 1) and $K$ has coordinates (3, -4). Then, $x_1 = -2$, $y_1 = 1$, $x_2 = 3$, and $y_2 = -4$. Plugging these values into the formula for the $y$-coordinate, we get:

$y = \frac{2(-4) + 3(1)}{5} = \frac{-8 + 3}{5} = \frac{-5}{5} = -1$

Therefore, the $y$-coordinate of the point that divides the line segment $JK$ in the ratio 2:3 is -1.

These examples demonstrate how the section formula can be applied to find the $y$-coordinate of a point dividing a line segment in a given ratio. The key is to correctly identify the coordinates of the endpoints and the ratio, and then substitute these values into the formula. The resulting value of $y$ represents the vertical position of the point that divides the line segment in the specified proportion.

Special Case: The Midpoint Formula

An interesting special case of the section formula arises when the ratio $m:n$ is equal to 1:1. In this scenario, the point $P$ divides the line segment $JK$ into two equal parts, effectively becoming the midpoint of the segment. Let's explore how the section formula simplifies to the midpoint formula in this particular case.

When $m = n$, we can substitute $m = 1$ and $n = 1$ into the section formula. The coordinates of the midpoint $P(x, y)$ are then given by:

$x = \frac{1 \cdot x_2 + 1 \cdot x_1}{1 + 1} = \frac{x_2 + x_1}{2}$

$y = \frac{1 \cdot y_2 + 1 \cdot y_1}{1 + 1} = \frac{y_2 + y_1}{2}$

These equations define the midpoint formula, which provides a straightforward method for calculating the coordinates of the midpoint of a line segment. The $x$-coordinate of the midpoint is simply the average of the $x$-coordinates of the endpoints, and the $y$-coordinate of the midpoint is the average of the $y$-coordinates of the endpoints. This intuitive result aligns perfectly with our understanding of the midpoint as the point that lies exactly halfway between the two endpoints.

The midpoint formula is a valuable tool in coordinate geometry, offering a quick and efficient way to find the center of a line segment. It has numerous applications in geometry, such as finding the center of a circle or the midpoint of a side of a triangle. The midpoint formula is also a fundamental concept in various mathematical fields, including calculus and linear algebra.

External Division

While we have primarily focused on internal division, where the point $P$ lies between the points $J$ and $K$, the section formula can also be extended to handle external division. In external division, the point $P$ lies on the line segment $JK$ extended beyond either point $J$ or point $K$. The ratio $m:n$ still represents the ratio of the distances $JP$ to $PK$, but in this case, one of the distances is considered negative.

To understand external division, let's consider the scenario where $P$ lies on the line segment $JK$ extended beyond point $K$. In this case, the distance $PK$ is considered negative because it is measured in the opposite direction from $JP$. The section formula for external division is similar to the formula for internal division, but with a crucial difference in the sign of $n$:

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

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

Notice that the only change is the replacement of the plus signs with minus signs in the numerators and the denominators. This sign change accounts for the fact that $P$ lies outside the segment $JK$.

Similarly, if $P$ lies on the line segment $JK$ extended beyond point $J$, the distance $JP$ is considered negative, and the section formula becomes:

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

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

External division can be a bit more challenging to visualize than internal division, but the section formula provides a consistent and reliable method for calculating the coordinates of the point $P$ in both cases. Understanding external division is crucial for a complete understanding of the section formula and its applications.

Conclusion

In conclusion, the section formula is a powerful and versatile tool in coordinate geometry that enables us to determine the coordinates of a point that divides a line segment in a given ratio. Whether the division is internal or external, the section formula provides a precise and elegant solution. We have explored the derivation of the formula, its application to finding the $y$-coordinate in a 2:3 ratio, and the special case of the midpoint formula. By mastering the section formula, you gain a fundamental tool for tackling a wide range of geometric problems and deepening your understanding of coordinate geometry. The section formula is not just a mathematical equation; it is a bridge connecting ratios and coordinates, enabling us to explore the geometry of lines and segments with greater precision and insight. This formula has practical applications in various fields, such as computer graphics, engineering, and physics, making it a valuable asset for students and professionals alike. Remember, the key to success with the section formula lies in understanding its derivation and practicing its application in various scenarios. So, embrace the power of the section formula and unlock the secrets of coordinate geometry.