To find the points of trisection of the line segment joining the points \( A(5, -6) \) and \( B(-3, 4) \), we will use the section formula. The points of trisection divide the line segment into three equal parts.
### Step 1: Identify the points and the ratios
Let \( A(5, -6) \) and \( B(-3, 4) \) be the two points. We need to find points \( P \) and \( Q \) such that \( AP = PQ = QB \). Since these segments are equal, we can say that:
- \( P \) divides \( AB \) in the ratio \( 1:2 \)
- \( Q \) divides \( AB \) in the ratio \( 2:1 \)
### Step 2: Find the coordinates of point \( P \)
Using the section formula, the coordinates of point \( P \) that divides \( AB \) in the ratio \( 1:2 \) are given by:
\[
P\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
where \( m = 1 \), \( n = 2 \), \( A(5, -6) = (x_1, y_1) \), and \( B(-3, 4) = (x_2, y_2) \).
Substituting the values:
\[
P\left( \frac{1 \cdot (-3) + 2 \cdot 5}{1 + 2}, \frac{1 \cdot 4 + 2 \cdot (-6)}{1 + 2} \right)
\]
Calculating the x-coordinate:
\[
P_x = \frac{-3 + 10}{3} = \frac{7}{3}
\]
Calculating the y-coordinate:
\[
P_y = \frac{4 - 12}{3} = \frac{-8}{3}
\]
Thus, the coordinates of point \( P \) are:
\[
P\left( \frac{7}{3}, \frac{-8}{3} \right)
\]
### Step 3: Find the coordinates of point \( Q \)
Using the section formula again, the coordinates of point \( Q \) that divides \( AB \) in the ratio \( 2:1 \) are given by:
\[
Q\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
where \( m = 2 \), \( n = 1 \).
Substituting the values:
\[
Q\left( \frac{2 \cdot (-3) + 1 \cdot 5}{2 + 1}, \frac{2 \cdot 4 + 1 \cdot (-6)}{2 + 1} \right)
\]
Calculating the x-coordinate:
\[
Q_x = \frac{-6 + 5}{3} = \frac{-1}{3}
\]
Calculating the y-coordinate:
\[
Q_y = \frac{8 - 6}{3} = \frac{2}{3}
\]
Thus, the coordinates of point \( Q \) are:
\[
Q\left( \frac{-1}{3}, \frac{2}{3} \right)
\]
### Final Answer
The points of trisection of the line segment joining the points \( (5, -6) \) and \( (-3, 4) \) are:
- \( P\left( \frac{7}{3}, \frac{-8}{3} \right) \)
- \( Q\left( \frac{-1}{3}, \frac{2}{3} \right) \)
