Find the points of trisection of line joining the points A (2, 1) and B (5, 3).

Answer

To find the points of trisection of the line joining the points A(2, 1) and B(5, 3), we will follow these steps:

### Step 1: Understand the concept of trisection
Trisection means dividing a line segment into three equal parts. Therefore, we need to find two points, P and Q, on the line segment AB such that AP = PQ = QB.

### Step 2: Determine the coordinates of points A and B
The coordinates of point A are (2, 1) and the coordinates of point B are (5, 3).

### Step 3: Calculate the ratio for trisection
For point P, which divides the line segment AB in the ratio 1:2 (AP:PB), and for point Q, which divides the line segment AB in the ratio 2:1 (AQ:QB).

### Step 4: Use the section formula
The section formula states that if a point divides a line segment joining points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the point (x, y) can be calculated as:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 5: Find the coordinates of point P
For point P (dividing AB in the ratio 1:2):
- \(m = 1\), \(n = 2\)
- \(x_1 = 2\), \(y_1 = 1\)
- \(x_2 = 5\), \(y_2 = 3\)

Using the section formula:
\[ x_P = \frac{1 \cdot 5 + 2 \cdot 2}{1 + 2} = \frac{5 + 4}{3} = \frac{9}{3} = 3 \]
\[ y_P = \frac{1 \cdot 3 + 2 \cdot 1}{1 + 2} = \frac{3 + 2}{3} = \frac{5}{3} \]

Thus, the coordinates of point P are \(P(3, \frac{5}{3})\).

### Step 6: Find the coordinates of point Q
For point Q (dividing AB in the ratio 2:1):
- \(m = 2\), \(n = 1\)

Using the section formula:
\[ x_Q = \frac{2 \cdot 5 + 1 \cdot 2}{2 + 1} = \frac{10 + 2}{3} = \frac{12}{3} = 4 \]
\[ y_Q = \frac{2 \cdot 3 + 1 \cdot 1}{2 + 1} = \frac{6 + 1}{3} = \frac{7}{3} \]

Thus, the coordinates of point Q are \(Q(4, \frac{7}{3})\).

### Final Answer
The points of trisection of the line segment joining A(2, 1) and B(5, 3) are:
- Point P: \( (3, \frac{5}{3}) \)
- Point Q: \( (4, \frac{7}{3}) \)

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