How to Find the Partition of a Line Segment in a Given Ratio?

Finding the partition of a line segment in a given ratio involves determining the precise coordinates of a point that divides the segment into two smaller segments with lengths proportional to the specified ratio. This is a fundamental concept in coordinate geometry, essential for various mathematical and practical applications.

Understanding the Ratio

When a line segment is partitioned in a ratio of a:b, it means that the point P divides the segment AB into two parts, AP and PB, such that the length of AP to the length of PB is a to b. The total number of parts the segment is divided into is a + b.

The Partition Formula

The coordinates of the point that partitions a line segment can be found using a specific formula, whether the segment is on a number line (one-dimensional) or in a coordinate plane (two-dimensional).

For a One-Dimensional Segment

If you have a segment on a number line with endpoints at x1 and x2, and you want to find the point x that partitions it in the ratio a:b, the formula is:

$$x = \frac{b \cdot x_1 + a \cdot x_2}{a + b}$$

For a Two-Dimensional Segment

For a segment in a coordinate plane with endpoints (x1, y1) and (x2, y2), and a ratio a:b, you calculate the x-coordinate and y-coordinate separately.

To find the x-coordinate of the partitioning point:
You will use the formula that takes the x-coordinates of the endpoints:
$$x = \frac{b \cdot x_1 + a \cdot x_2}{a + b}$$
To find the y-coordinate of the partitioning point:
Similarly, for the y-coordinate, you use the y-coordinates of the endpoints:
$$y = \frac{b \cdot y_1 + a \cdot y_2}{a + b}$$

The point that partitions the segment will then be (x, y).

Step-by-Step Guide to Partitioning a Segment

To find the point that partitions a segment in a given ratio, follow these steps:

Identify the Endpoints: Clearly label the coordinates of the segment's endpoints. Let the first endpoint be (x1, y1) and the second endpoint be (x2, y2).
Determine the Ratio: Identify the given ratio a:b. Here, a represents the first part of the ratio and b represents the second part.
Apply the X-coordinate Formula: Substitute the values of x1, x2, a, and b into the formula for the x-coordinate:
$$x = \frac{(b \times x_1) + (a \times x_2)}{a + b}$$
Apply the Y-coordinate Formula: Substitute the values of y1, y2, a, and b into the formula for the y-coordinate:
$$y = \frac{(b \times y_1) + (a \times y_2)}{a + b}$$
State the Partitioning Point: The calculated x and y values give you the coordinates (x, y) of the point that partitions the segment in the desired ratio.

Example Calculation

Let's find the point P that partitions the line segment with endpoints A(2, 5) and B(8, 17) in the ratio 1:2.

Endpoints: (x1, y1) = (2, 5) and (x2, y2) = (8, 17).
Ratio: a:b = 1:2, so a = 1 and b = 2.
Calculate X-coordinate:
$$x = \frac{(2 \times 2) + (1 \times 8)}{1 + 2}$$
$$x = \frac{4 + 8}{3}$$
$$x = \frac{12}{3}$$
$$x = 4$$
Calculate Y-coordinate:
$$y = \frac{(2 \times 5) + (1 \times 17)}{1 + 2}$$
$$y = \frac{10 + 17}{3}$$
$$y = \frac{27}{3}$$
$$y = 9$$
Partitioning Point: The point P that partitions the segment AB in the ratio 1:2 is (4, 9).

Practical Applications

Partitioning a line segment is a concept frequently used in various fields:

Geometry: For dividing shapes, finding centroids, or determining specific points on lines.
Computer Graphics: Used in algorithms for interpolation, rendering curves (e.g., Bézier curves), and scaling objects.
Physics: In calculating centers of mass or weighted averages.
Engineering: In design and analysis, such as locating specific points on structural elements.

Understanding this method provides a foundational skill for more complex geometric and algebraic problems.