Find the coordinates of the point of trisection

General
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find the coordinates of the point of trisection

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Find the coordinates of the point of trisection

Answer: To find the coordinates of the points of trisection of a 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 as ( A(x_1, y_1) ) and ( B(x_2, y_2) ).

The points of trisection will divide the segment ( AB ) into three equal parts. Let’s call these points ( P ) and ( Q ), where ( P ) is closer to ( A ) and ( Q ) is closer to ( B ).

Steps to Find the Coordinates:

  1. Find the coordinates of point ( P ):

    • Point ( P ) divides the segment ( AB ) in the ratio ( 1:2 ).
    • The formula for the coordinates of a point dividing a line segment in the ratio ( m:n ) is:
      \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
    • For ( P ), the ratio is ( 1:2 ):
      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) = \left( \frac{x_2 + 2x_1}{3}, \frac{y_2 + 2y_1}{3} \right)

  2. Find the coordinates of point ( Q ):

    • Point ( Q ) divides the segment ( AB ) in the ratio ( 2:1 ).
    • Using the same formula for ( Q ):
      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) = \left( \frac{2x_2 + x_1}{3}, \frac{2y_2 + y_1}{3} \right)

Example:

Let’s use an example to illustrate this process. Suppose we have points ( A(2, 3) ) and ( B(8, 9) ).

  1. Coordinates of ( P ):

    P \left( \frac{8 + 2 \cdot 2}{3}, \frac{9 + 2 \cdot 3}{3} \right) = \left( \frac{8 + 4}{3}, \frac{9 + 6}{3} \right) = \left( \frac{12}{3}, \frac{15}{3} \right) = (4, 5)

  2. Coordinates of ( Q ):

    Q \left( \frac{2 \cdot 8 + 2}{3}, \frac{2 \cdot 9 + 3}{3} \right) = \left( \frac{16 + 2}{3}, \frac{18 + 3}{3} \right) = \left( \frac{18}{3}, \frac{21}{3} \right) = (6, 7)

Conclusion:

Therefore, the coordinates of the points of trisection of the line segment joining ( A(2, 3) ) and ( B(8, 9) ) are ( P(4, 5) ) and ( Q(6, 7) ).