Question

Find the coordinates of the points of trisection of the line segment joining the points \( A(-1, 4) \) and \( B(-3, -2) \).

Hint:

The trisection points are also the midpoints of segments formed by other trisection points. For example, Q is the midpoint of PB.

Answer 1

Step 1: Understanding the Concept:
Trisection divides a line segment into three equal parts. Two points, P and Q, divide AB in the ratios 1:2 and 2:1 respectively.
Step 2: Detailed Explanation:
Point P divides AB in ratio \( 1:2 \):
\( x = \frac{1(-3) + 2(-1)}{1 + 2} = \frac{-5}{3} \).
\( y = \frac{1(-2) + 2(4)}{1 + 2} = \frac{6}{3} = 2 \).
So, \( P = (-5/3, 2) \).
Point Q divides AB in ratio \( 2:1 \):
\( x = \frac{2(-3) + 1(-1)}{2 + 1} = \frac{-7}{3} \).
\( y = \frac{2(-2) + 1(4)}{2 + 1} = \frac{0}{3} = 0 \).
So, \( Q = (-7/3, 0) \).
Step 3: Final Answer:
The trisection points are \( (-5/3, 2) \) and \( (-7/3, 0) \).