Line Segment Division

What is Line Segment Division?

Dividing a line segment involves identifying a point that divides the segment either internally or externally in a specific ratio. The line segment is typically defined by its endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$.

Division Formula

If a point $P(x, y)$ divides the segment $AB$ in the ratio $m : n$, the coordinates of $P$ are given by:

Internally:

$$P(x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)$$

Externally:

$$P(x, y) = \left( \frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n} \right)$$

Special Case: Midpoint

When $m = n$, the point $P$ is the midpoint of the segment, and the formula simplifies to:

$$P(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Applications

• Finding a point that divides a segment in a specific ratio.
• Solving problems involving proportional relationships.
• Identifying midpoints for symmetry or geometric constructions.

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Worked Examples

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Summary

• The division formula calculates the coordinates of a point dividing a segment in a given ratio.
• Internal Division:

$$P(x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)$$

• External Division:

$$P(x, y) = \left( \frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n} \right)$$

• Special case: midpoint occurs when $m = n$
• Useful for geometry, proportionality problems, and graphing.