If the vectors `vec(AB)=3hati+4hatk and vec(AC)=5hati-2hatj+4hatk` are the sides of a triangle ABC, then the length of the median through A is (A) `sqrt(33)` (B) `sqrt(45)` (C) `sqrt(18)` (D) `sqrt(720`

Options

A. `sqrt(18)`
B. `sqrt(72)`
C. `sqrt(33)`
D. `sqrt(45)`

Answer

To find the length of the median through point A in triangle ABC, we will follow these steps:

### Step 1: Identify the vectors
We are given:
- \(\vec{AB} = 3\hat{i} + 4\hat{k}\)
- \(\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}\)

### Step 2: Find the coordinates of points B and C
Assuming point A is at the origin (0, 0, 0):
- Point B can be represented as \(B(3, 0, 4)\)
- Point C can be represented as \(C(5, -2, 4)\)

### Step 3: Find the midpoint D of segment BC
The coordinates of the midpoint D can be calculated using the midpoint formula:
\[ D = \left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2}\right) \]
Substituting the coordinates of B and C:
\[ D = \left(\frac{3 + 5}{2}, \frac{0 - 2}{2}, \frac{4 + 4}{2}\right) = \left(\frac{8}{2}, \frac{-2}{2}, \frac{8}{2}\right) = (4, -1, 4) \]

### Step 4: Find the vector \(\vec{AD}\)
The vector \(\vec{AD}\) can be calculated as:
\[ \vec{AD} = D - A = (4, -1, 4) - (0, 0, 0) = (4, -1, 4) \]
Thus, \(\vec{AD} = 4\hat{i} - 1\hat{j} + 4\hat{k}\).

### Step 5: Calculate the magnitude of \(\vec{AD}\)
The magnitude of the vector \(\vec{AD}\) is given by:
\[ |\vec{AD}| = \sqrt{(4)^2 + (-1)^2 + (4)^2} = \sqrt{16 + 1 + 16} = \sqrt{33} \]

### Conclusion
The length of the median through point A is \(\sqrt{33}\).