Question

If the sides $AB$ and $AC$ of $\triangle ABC$ are represented by vectors $\hat{i} + \hat{j} + 4 \hat{k}$ and $3 \hat{i} - \hat{j} + 4 \hat{k}$ respectively, then the length of the median through A on BC is :

• A. $2 \sqrt{2}$ units
• B. $\sqrt{18}$ units
• C. $\frac{\sqrt{34}}{2}$ units
• D. $\frac{\sqrt{48}}{2}$ units

Hint:

To find the length of a median, find the midpoint of the opposite side and calculate the distance between the vertex and the midpoint.

Answer 1

The Correct Option is C

The median's length is the distance between point A and the midpoint of segment BC. The midpoint of BC is calculated by averaging the coordinates of B $(1, 1, 4)$ and C $(3, -1, 4)$. This midpoint, M, is determined as: \[\mathbf{M} = \left( \frac{1 + 3}{2}, \frac{1 + (-1)}{2}, \frac{4 + 4}{2} \right) = (2, 0, 4).\] The distance from A $(\hat{i} + \hat{j} + 4\hat{k})$ to M $(2, 0, 4)$ is: \[\d = \sqrt{(2 - 1)^2 + (0 - 1)^2 + (4 - 4)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{2}.\] Consequently, the length of the median is $\frac{\sqrt{34}}{2}$.