The points of trisection of the segment joining the points (1, 0, 2) and (1, 3, 2) are:
A. \(\left(1, \displaystyle\frac{3}{2}, \frac{4}{3}\right)\)
B. (1, 1, 2)
C. (1, 2, 2)
D. \(\left(1, \displaystyle\frac{3}{2}, 2\right)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and C only
The points of trisection divide the segment from P(1, 0, 2) to Q(1, 3, 2) in the ratios 1 : 2 and 2 : 1. By the section formula:
\( \displaystyle \text{ratio } 1:2: \quad \left(\frac{1(1)+2(1)}{3}, \frac{1(3)+2(0)}{3}, \frac{1(2)+2(2)}{3}\right) = (1, 1, 2) \)
\( \displaystyle \text{ratio } 2:1: \quad \left(\frac{2(1)+1(1)}{3}, \frac{2(3)+1(0)}{3}, \frac{2(2)+1(2)}{3}\right) = (1, 2, 2) \)
So the points of trisection are (1, 1, 2) and (1, 2, 2).
Checking the statements
(A) \(\left(1, \frac{3}{2}, \frac{4}{3}\right)\) — Incorrect. Every point of the segment has z = 2, not \(\frac{4}{3}\).
(B) (1, 1, 2) — Correct. It divides PQ in the ratio 1 : 2.
(C) (1, 2, 2) — Correct. It divides PQ in the ratio 2 : 1.
(D) \(\left(1, \frac{3}{2}, 2\right)\) — Incorrect. This is the mid-point, ratio 1 : 1.