Question

The coordinates of the points of trisection of the line segment joining the points $ (3, 2) $ and $ (6, -4) $ are

• A. $ (4, 0) $ and $ (5, -2) $
• B. $ (-1, 0) $ and $ (-4, 2) $
• C. $ \left( \frac{11}{3}, 0 \right) $ and $ \left( \frac{13}{3}, -2 \right) $
• D. $ (0, 1) $ and $ (2, 3) $

Hint:

After finding one trisection point, the second can also be found quickly as the midpoint between that point and the second endpoint.

Answer 1

The Correct Option is A

Concept: Trisection means dividing a line segment into three equal parts. If a segment $AB$ is divided into three equal parts, then two points are formed:
• First point divides $AB$ in ratio $1:2$
• Second point divides $AB$ in ratio $2:1$ We use the Section Formula: \[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Step 1: Identify endpoints.
\[ A(3,2), \quad B(6,-4) \]

Step 2: Find first trisection point (1:2).
\[ x = \frac{1\cdot 6 + 2\cdot 3}{3} = \frac{6+6}{3} = 4 \] \[ y = \frac{1\cdot (-4) + 2\cdot 2}{3} = \frac{-4+4}{3} = 0 \] So first point: \[ P = (4,0) \]

Step 3: Find second trisection point (2:1).
\[ x = \frac{2\cdot 6 + 1\cdot 3}{3} = \frac{12+3}{3} = 5 \] \[ y = \frac{2\cdot (-4) + 1\cdot 2}{3} = \frac{-8+2}{3} = -2 \] So second point: \[ Q = (5,-2) \]

Step 4: Conclusion.
The points of trisection are: \[ \boxed{(4,0) \text{ and } (5,-2)} \]