How to Find the Ratio in Which a Point Divides a Line Segment Internally and Externally

Do you need to determine the ratio in which a point divides a given line segment? This article will guide you through the process of finding this ratio both internally and externally. By the end, you'll be able to use the section formula to solve similar problems effectively.

Internal Division

In geometry, the section formula helps us find the coordinates of a point that divides a line segment in a given ratio. When a point P(x, y) divides the line segment joining points A(x_1, y_1) and B(x_2, y_2) internally in the ratio m:n, the coordinates of P can be calculated using the following equations:

x (mx_2 nx_1) / (m n)

y (my_2 ny_1) / (m n)

Let's apply these formulas to find the internal division ratio for the specific points given in the example:

Example: Finding the Internal Division Ratio

Given points:

A(1, 2) B(6, 7) P(3, 4)

Our goal is to find the ratio m:n such that P(3, 4) divides the line segment AB internally.

For the x-coordinates:

3 (6m 1n) / (m n)

Multiplying through by (m n)

3(m n) 6m n

Rearranging gives:

3n - n 6m - 3m

2n 3m

n/m 3/2

n:m 3:2

For the y-coordinates:

4 (7m 2n) / (m n)

Multiplying through by (m n)

4(m n) 7m 2n

Rearranging gives:

4n - 2n 7m - 4m

2n 3m

n/m 3/2

n:m 3:2

Thus, the point P(3, 4) divides the segment AB internally in the ratio 3:2.

External Division

When a point P(x, y) divides the line segment joining points A(x_1, y_1) and B(x_2, y_2) externally in the ratio m:n, the coordinates of P can be determined using the following modified formulas:

x (mx_2 - nx_1) / (m - n)

y (my_2 - ny_1) / (m - n)

Example: Finding the External Division Ratio

Using the same points as before, let's determine the external division ratio.

For the x-coordinates:

3 (6m - 1n) / (m - n)

Multiplying through by (m - n)

3(m - n) 6m - n

Rearranging gives:

-3n - n 6m - 3m

-2n 3m

n/m -3/2

n:m -3:2

For the y-coordinates:

4 (7m - 2n) / (m - n)

Multiplying through by (m - n)

4(m - n) 7m - 2n

Rearranging gives:

-4n - 2n 7m - 4m

-2n 3m

n/m -3/2

n:m -3:2

Therefore, the point P(3, 4) divides the segment AB externally in the ratio -3:2.

Summary

The finding of the division ratio can be summarized as follows:

Internal Division Ratio: 3:2

External Division Ratio: -3:2

Understanding both internal and external division is crucial in solving geometric problems and finding the location of points on a line segment. By using the section formula, you can easily determine the division ratio of any point on a given segment.