Find the coordinates of the points that trisect the line segment joining (1, - 2) and (-3, 4)

Options

A. `((-1)/(3),0)`
B. `((-5)/(3),2)`
C. Both (a) and (b)
D. None of these

Answer

To find the coordinates of the points that trisect the line segment joining the points \( A(1, -2) \) and \( B(-3, 4) \), we will follow these steps:

### Step 1: Understand Trisection
Trisection means dividing the line segment into three equal parts. Therefore, we need to find two points, \( C \) and \( D \), such that \( AC = CD = DB \).

### Step 2: Use the Section Formula
The section formula for finding the coordinates of a point dividing a line segment in the ratio \( m:n \) is given by:
\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \).

### Step 3: Find Coordinates of Point \( C \)
Point \( C \) divides \( AB \) in the ratio \( 1:2 \) (1 part from \( A \) and 2 parts towards \( B \)).

- Here, \( m = 1 \), \( n = 2 \), \( A(1, -2) \) is \( (x_1, y_1) \), and \( B(-3, 4) \) is \( (x_2, y_2) \).

Using the section formula:
\[ x_C = \frac{1 \cdot (-3) + 2 \cdot 1}{1 + 2} = \frac{-3 + 2}{3} = \frac{-1}{3} \]
\[ y_C = \frac{1 \cdot 4 + 2 \cdot (-2)}{1 + 2} = \frac{4 - 4}{3} = 0 \]
Thus, the coordinates of point \( C \) are \( \left( -\frac{1}{3}, 0 \right) \).

### Step 4: Find Coordinates of Point \( D \)
Point \( D \) divides \( AB \) in the ratio \( 2:1 \) (2 parts from \( A \) and 1 part towards \( B \)).

- Here, \( m = 2 \), \( n = 1 \).

Using the section formula:
\[ x_D = \frac{2 \cdot (-3) + 1 \cdot 1}{2 + 1} = \frac{-6 + 1}{3} = \frac{-5}{3} \]
\[ y_D = \frac{2 \cdot 4 + 1 \cdot (-2)}{2 + 1} = \frac{8 - 2}{3} = \frac{6}{3} = 2 \]
Thus, the coordinates of point \( D \) are \( \left( -\frac{5}{3}, 2 \right) \).

### Final Answer
The coordinates of the points that trisect the line segment joining \( (1, -2) \) and \( (-3, 4) \) are:
- Point \( C \): \( \left( -\frac{1}{3}, 0 \right) \)
- Point \( D \): \( \left( -\frac{5}{3}, 2 \right) \)