The vertices of a triangle are A(1, 2, 0), B(3, 3, 2) and C(3, 5, 6), and M is the midpoint of BC. Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Line | Direction cosines | ||
| (A) | AB | (I) | \(\displaystyle 0, \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\) |
| (B) | AC | (II) | \(\displaystyle \frac{2}{7}, \frac{3}{7}, \frac{6}{7}\) |
| (C) | BC | (III) | \(\displaystyle \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}\) |
| (D) | AM | (IV) | \(\displaystyle \frac{2}{3}, \frac{1}{3}, \frac{2}{3}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
For each line, the direction ratios are the differences of the coordinates of its end points; dividing by their length gives the direction cosines.
(A) AB
This matches (IV) \(\frac{2}{3}, \frac{1}{3}, \frac{2}{3}\):
\(3-1,\ 3-2,\ 2-0 = 2, 1, 2\), and \(\sqrt{4+1+4} = 3\)
(B) AC
This matches (II) \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\):
\(3-1,\ 5-2,\ 6-0 = 2, 3, 6\), and \(\sqrt{4+9+36} = 7\)
(C) BC
This matches (I) \(0, \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\):
\(3-3,\ 5-3,\ 6-2 = 0, 2, 4\), and \(\sqrt{0+4+16} = 2\sqrt{5}\)
(D) AM
This matches (III) \(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}\). The midpoint of BC is M(3, 4, 4):
\(3-1,\ 4-2,\ 4-0 = 2, 2, 4\), and \(\sqrt{4+4+16} = 2\sqrt{6}\)