Division of a Line Segment

Division of a Line Segment#

Distance Formula Revision#

Theory#

Before exploring line segment division, let’s review the distance formula:

  • The distance between points \((x_1, y_1)\) and \((x_2, y_2)\) is: $\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)$

  • This forms the foundation for finding points that divide line segments

Application#

If \(A(2, 3)\) and \(B(8, 7)\):

  • Distance \(AB = \sqrt{(8-2)^2 + (7-3)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}\)

Division of Line Segments#

Theory#

A line segment can be divided internally or externally in a given ratio.

Internal Division#

If point \(P\) divides line segment \(AB\) internally in the ratio \(m:n\), then: $\(P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)\)$

where \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the endpoints.

External Division#

If point \(P\) divides line segment \(AB\) externally in the ratio \(m:n\), then: $\(P = \left(\frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n}\right)\)$

Note: External division is only possible when \(m \neq n\).

Special Case: Midpoint#

When \(m = n = 1\), we get the midpoint formula: $\(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)$

Interactive Visualization: Line Segment Division Explorer#

Application#

Examples#

Example 1: Internal Division

Find the point that divides the line segment joining \(A(2, 3)\) and \(B(8, 11)\) internally in the ratio \(2:3\).

Solution: Using the internal division formula with \(m = 2\), \(n = 3\): $\(P = \left(\frac{2(8) + 3(2)}{2 + 3}, \frac{2(11) + 3(3)}{2 + 3}\right)\)\( \)\(P = \left(\frac{16 + 6}{5}, \frac{22 + 9}{5}\right)\)\( \)\(P = \left(\frac{22}{5}, \frac{31}{5}\right) = (4.4, 6.2)\)$

Example 2: Finding the Ratio

Point \(P(5, 7)\) divides the line segment joining \(A(3, 3)\) and \(B(6, 9)\). Find the ratio.

Solution: Let the ratio be \(m:n\). Using the internal division formula: $\(5 = \frac{m(6) + n(3)}{m + n}\)\( \)\(5(m + n) = 6m + 3n\)\( \)\(5m + 5n = 6m + 3n\)\( \)\(2n = m\)$

Therefore, the ratio is \(m:n = 2:1\).

Example 3: External Division

Find the point that divides the line segment joining \(A(1, 2)\) and \(B(5, 8)\) externally in the ratio \(3:1\).

Solution: Using the external division formula with \(m = 3\), \(n = 1\): $\(P = \left(\frac{3(5) - 1(1)}{3 - 1}, \frac{3(8) - 1(2)}{3 - 1}\right)\)\( \)\(P = \left(\frac{15 - 1}{2}, \frac{24 - 2}{2}\right)\)\( \)\(P = \left(\frac{14}{2}, \frac{22}{2}\right) = (7, 11)\)$

Multiple Choice Questions#

Sector Specific Questions: Line Segment Division Applications#

Key Takeaways#

Important

Division of Line Segments - Essential Concepts

  1. Internal Division Formula: For ratio \(m:n\), point \(P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)\)

  2. External Division Formula: For ratio \(m:n\) (where \(m \neq n\)), point \(P = \left(\frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n}\right)\)

  3. Special Cases:

    • Midpoint: When \(m = n\), use \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

    • Trisection: Divide in ratios \(1:2\) and \(2:1\) for three equal parts

    • Golden ratio: Use \(\phi:1\) where \(\phi \approx 1.618\)

  4. Applications:

    • Engineering: Structural support placement

    • Science: Data point interpolation

    • Finance: Risk-return balancing

    • Art: Compositional harmony

  5. Key Distinction: Internal division creates a point between the endpoints, while external division creates a point beyond one endpoint

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