The vertices of a triangle are A(2, −1, 3), B(4, 1, 1) and C(2, 5, −1). The Cartesian equation of the median through A is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{x-2}{1}=\frac{y+1}{4}=\frac{z-3}{-3}\)
The median through A joins A to the midpoint D of BC:
\( \displaystyle D = \left(\frac{4+2}{2}, \frac{1+5}{2}, \frac{1-1}{2}\right) = (3, 3, 0) \)
Direction ratios of AD:
\(3-2,\ 3-(-1),\ 0-3 = 1, 4, -3\)
The line through A(2, −1, 3) with these direction ratios is
\( \displaystyle \frac{x-2}{1}=\frac{y+1}{4}=\frac{z-3}{-3} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{x-2}{-1}=\frac{y+1}{2}=\frac{z-3}{-1}\) — Incorrect. Its direction −1, 2, −1 is that of BC; this is the line through A parallel to BC.
Option (3) \(\displaystyle\frac{x-2}{4}=\frac{y+1}{7}=\frac{z-3}{-3}\) — Incorrect. This uses B + C = (6, 6, 0) in place of the midpoint, without halving.
Option (4) \(\displaystyle\frac{x-2}{1}=\frac{y+1}{4}=\frac{z-3}{3}\) — Incorrect. The z-ratio is \(0-3 = -3\), not 3.