Dividing the Line Segment: The Point -124/5 and Its Ratio

In this article, we explore the division of a line segment in a specific ratio using the example where the point -124/5 divides the line joining points 14 and -316. We will delve into the mathematical details to understand the calculations involved.

Introduction to Line Segment Division

The concept of dividing a line segment into a specific ratio is a fundamental topic in geometry and has practical applications in various fields, including computer graphics, architecture, and engineering. In this article, we will focus on the division of the line segment joining points 14 and -316 by the point -124/5.

The Problem: Point -124/5 and Its Division

Consider the points A with coordinates 14 and B with coordinates -316. Let P be the point -124/5. We need to determine the ratio in which P divides the line segment AB.

Calculating the Distances AP and PB

Let's start by calculating the distance AP and PB. The formula for the distance between two points in a coordinate system can be used here.

Distance AP

AP is calculated as follows:

AP |14 - (-124/5)| |14 124/5| |70/5 124/5| |194/5| 194/5.

Distance PB

PB is calculated as follows:

PB |-316 - (-124/5)| |-316 124/5| |-1580/5 124/5| |-1456/5| 1456/5.

Calculating the Ratio AP/PB

The ratio AP/PB can be calculated as follows:

AP/PB (194/5) / (1456/5) 194/1456

Further simplifying this ratio:

AP/PB 194/1456 97/728

An Alternative Approach

Another approach to verify this can be through the midpoint theorem. We can check if -124/5 is a midpoint of the line segment AB.

Checking if -124/5 is the Midpoint

To be a midpoint, the point must be equidistant from A and B. Let's calculate the midpoints mathematically:

The midpoint formula is given by ((x1 x2)/2, (y1 y2)/2)

Here, the x-coordinates are 14 and -316.

Midpoint ((14 (-316))/2) (-302/2) -151.

This shows that -124/5 is not the midpoint, as -151 ≠ -124/5.

Conclusion

In conclusion, the point -124/5 divides the line segment joining points 14 and -316 in the ratio 97/728. This division is not in the same ratio as the midpoint, which further confirms that -124/5 does not divide the line segment at a midpoint position.

Understanding such geometric divisions is crucial for various applications and helps in solving complex problems in mathematics and its related fields.