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Mar 1, 2025 by ADMIN 189 views

What are the $x$ - and $y$ -coordinates of point $E$ , which partitions the directed line segment from $A$ to $B$ into a ratio of $1:2$ ?

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 partitioned into a ratio, it means that the line segment is divided into two or more parts, with each part having a specific length or proportion. In this article, we will explore how to find the $x$ - and $y$ -coordinates of a point that partitions a directed line segment into a ratio of $1:2$ .

To solve this problem, we need to understand the concept of ratios and how they apply to line segments. A ratio of $1:2$ means that the line segment is divided into two parts, with the first part having a length of $1$ unit and the second part having a length of $2$ units. This means that the point $E$ will be located at a position that is $1/3$ of the way from point $A$ to point $B$ .

The section formula is a mathematical formula that allows us to find the coordinates of a point that divides a line segment into a ratio. The section formula is given by:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two end points of the line segment, and $m:n$ is the ratio in which the line segment is divided.

In this problem, we are given that the line segment is divided into a ratio of $1:2$ , which means that $m:n = 1:2$ . We are also given the coordinates of the two end points, $A$ and $B$ . Let's assume that the coordinates of $A$ are $(x_1, y_1)$ and the coordinates of $B$ are $(x_2, y_2)$ .

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

\left( \frac{1(x_2) + 2(x_1)}{1+2}, \frac{1(y_2) + 2(y_1)}{1+2} \right)

Simplifying the expression, we get:

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

Now that we have the formula for finding the coordinates of point $E$ , we can plug in the values of $x_1$ , $y_1$ , $x_2$ , and $y_2$ to find the coordinates of point $E$ .

Let's assume that the coordinates of $A$ are $(2, 3)$ and the coordinates of $B$ are $(6, 9)$ . Plugging in these values, we get:

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

Simplifying the expression, we get:

\left( \frac{6 + 4}{3}, \frac{9 + 6}{3} \right)

\left( \frac{10}{3}, \frac{15}{3} \right)

\left( \frac{10}{3}, 5 \right)

Therefore, the coordinates of point $E$ are $\left( \frac{10}{3}, 5 \right)$ .

In this article, we explored how to find the $x$ - and $y$ -coordinates of a point that partitions a directed line segment into a ratio of $1:2$ . We used the section formula to find the coordinates of point $E$ , and we applied the formula to a specific example to find the coordinates of point $E$ . We hope that this article has provided a clear understanding of how to solve this type of problem.

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Q&A: What are the $x$ - and $y$ -coordinates of point $E$ , which partitions the directed line segment from $A$ to $B$ into a ratio of $1:2$ ?

In our previous article, we explored how to find the $x$ - and $y$ -coordinates of a point that partitions a directed line segment into a ratio of $1:2$ . We used the section formula to find the coordinates of point $E$ , and we applied the formula to a specific example to find the coordinates of point $E$ . In this article, we will answer some frequently asked questions related to this topic.

A: The section formula is a mathematical formula that allows us to find the coordinates of a point that divides a line segment into a ratio. The section formula is given by:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two end points of the line segment, and $m:n$ is the ratio in which the line segment is divided.

A: To apply the section formula, you need to know the coordinates of the two end points of the line segment, $A$ and $B$ , and the ratio in which the line segment is divided. Let's assume that the coordinates of $A$ are $(x_1, y_1)$ and the coordinates of $B$ are $(x_2, y_2)$ . Using the section formula, you can find the coordinates of point $E$ as follows:

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

A: If the ratio is not $1:2$ , you can still use the section formula to find the coordinates of point $E$ . Let's assume that the ratio is $a:b$ . Using the section formula, you can find the coordinates of point $E$ as follows:

\left( \frac{ax_2 + bx_1}{a+b}, \frac{ay_2 + by_1}{a+b} \right)

A: Yes, you can use the section formula to find the coordinates of point $E$ even if the line segment is not directed. The section formula works for both directed and undirected line segments.

A: Yes, you can still find the coordinates of point $E$ even if you don't know the coordinates of point $A$ or point $B$ . You can use the section formula to find the coordinates of point $E$ in terms of the coordinates of point $A$ and point $B$ .

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

In this article, we answered some frequently asked questions related to finding the $x$ - and $y$ -coordinates of a point that partitions a directed line segment into a ratio of $1:2$ . We hope that this article has provided a clear understanding of how to solve this type of problem.