Find the coordinates of the point of trisection of the line segment joining

General
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find the coordinates of the point of trisection of the line segment joining

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Find the coordinates of the point of trisection of the line segment joining

Answer: To find the coordinates of the points of trisection of the line segment joining two points, we need to divide the line segment into three equal parts. Let’s denote the endpoints of the line segment by A(x_1, y_1) and B(x_2, y_2).

Step-by-Step Solution:

  1. Identify the coordinates of the endpoints:

    • Let A(x_1, y_1) and B(x_2, y_2) be the endpoints of the line segment.

  2. Find the points of trisection:

    • To trisect the line segment, we need to find two points, P and Q, such that P is one-third of the way from A to B and Q is two-thirds of the way from A to B.

  3. Use the section formula:

    • The section formula for dividing a line segment in the ratio m:n is given by:
      \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)

  4. First Point of Trisection (P):

    • The point P divides the line segment in the ratio 1:2.
    • Using the section formula:
      P = \left(\frac{1 \cdot x_2 + 2 \cdot x_1}{1+2}, \frac{1 \cdot y_2 + 2 \cdot y_1}{1+2}\right)
    • Simplifying, we get:
      P = \left(\frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3}\right)

  5. Second Point of Trisection (Q):

    • The point Q divides the line segment in the ratio 2:1.
    • Using the section formula:
      Q = \left(\frac{2 \cdot x_2 + 1 \cdot x_1}{2+1}, \frac{2 \cdot y_2 + 1 \cdot y_1}{2+1}\right)
    • Simplifying, we get:
      Q = \left(\frac{2x_2 + x_1}{3}, \frac{2y_2 + y_1}{3}\right)

Final Coordinates:

  • The coordinates of the points of trisection of the line segment joining A(x_1, y_1) and B(x_2, y_2) are:
    • First Point of Trisection P: \left(\frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3}\right)
    • Second Point of Trisection Q: \left(\frac{2x_2 + x_1}{3}, \frac{2y_2 + y_1}{3}\right)

These points divide the line segment into three equal parts.