To find the coordinates of the points of trisection of the line segment joining the points A(7, -2) and B(1, 5), we will follow these steps:
### Step 1: Identify the Points
We have two points:
- Point A: \( A(7, -2) \)
- Point B: \( B(1, 5) \)
### Step 2: Understand Trisection
Trisection means dividing the line segment into three equal parts. This will give us two points of trisection, which we will denote as P and P'.
### Step 3: Use the Section Formula
The coordinates of a point dividing a line segment in the ratio \( m:n \) can be calculated using the section formula:
\[
\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points A and B respectively.
### Step 4: Calculate Coordinates of Point P
For the first point of trisection (P), we divide the segment in the ratio \( 1:2 \) (1 part towards A and 2 parts towards B).
Here:
- \( m = 1 \)
- \( n = 2 \)
- \( (x_1, y_1) = (7, -2) \)
- \( (x_2, y_2) = (1, 5) \)
Using the section formula:
\[
P = \left( \frac{1 \cdot 1 + 2 \cdot 7}{1 + 2}, \frac{1 \cdot 5 + 2 \cdot (-2)}{1 + 2} \right)
\]
Calculating the x-coordinate:
\[
x_P = \frac{1 + 14}{3} = \frac{15}{3} = 5
\]
Calculating the y-coordinate:
\[
y_P = \frac{5 - 4}{3} = \frac{1}{3}
\]
Thus, the coordinates of point P are \( P(5, \frac{1}{3}) \).
### Step 5: Calculate Coordinates of Point P'
For the second point of trisection (P'), we divide the segment in the ratio \( 2:1 \) (2 parts towards A and 1 part towards B).
Here:
- \( m = 2 \)
- \( n = 1 \)
Using the section formula again:
\[
P' = \left( \frac{2 \cdot 1 + 1 \cdot 7}{2 + 1}, \frac{2 \cdot 5 + 1 \cdot (-2)}{2 + 1} \right)
\]
Calculating the x-coordinate:
\[
x_{P'} = \frac{2 + 7}{3} = \frac{9}{3} = 3
\]
Calculating the y-coordinate:
\[
y_{P'} = \frac{10 - 2}{3} = \frac{8}{3}
\]
Thus, the coordinates of point P' are \( P'(3, \frac{8}{3}) \).
### Final Answer
The coordinates of the points of trisection are:
- Point P: \( (5, \frac{1}{3}) \)
- Point P': \( (3, \frac{8}{3}) \)
