Show that P (3,m -5) is a point of trisection of the line segment joining the points A (4,-2) and B (1, 4). Hence, find the value of 'm'.

Answer

To show that the point \( P(3, m - 5) \) is a point of trisection of the line segment joining the points \( A(4, -2) \) and \( B(1, 4) \), we will use the section formula and find the value of \( m \).

### Step 1: Understand the concept of trisection
A point of trisection divides the line segment into three equal parts. Therefore, if \( P \) is a point of trisection of \( AB \), it divides the segment \( AB \) in the ratio \( 1:2 \) or \( 2:1 \).

### Step 2: Use the section formula
The coordinates of a point \( P(x, y) \) that divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) can be calculated using the section formula:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 3: Apply the section formula for the x-coordinate
Given \( A(4, -2) \) and \( B(1, 4) \), and \( P(3, m - 5) \) dividing \( AB \) in the ratio \( 1:2 \):
- \( m = 1 \)
- \( n = 2 \)

Using the section formula for the x-coordinate:
\[ 3 = \frac{1 \cdot 1 + 2 \cdot 4}{1 + 2} \]
Calculating the right-hand side:
\[ 3 = \frac{1 + 8}{3} = \frac{9}{3} = 3 \]
This confirms that the x-coordinate is correct.

### Step 4: Apply the section formula for the y-coordinate
Now, we will calculate the y-coordinate:
\[ m - 5 = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} \]
Calculating the right-hand side:
\[ m - 5 = \frac{4 - 4}{3} = \frac{0}{3} = 0 \]
Thus, we have:
\[ m - 5 = 0 \]

### Step 5: Solve for \( m \)
To find \( m \):
\[ m = 5 \]

### Conclusion
Thus, we have shown that \( P(3, m - 5) \) is a point of trisection of the line segment joining \( A(4, -2) \) and \( B(1, 4) \), and the value of \( m \) is:
\[ \boxed{5} \]