If the vectors vec AB = 3hat i + 4hat k and vec AC = 5hat i - 2hat j + 4hat k are the sides of a triangle ABC, then the length of the median through A is:

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EEJ MAIN Mathematics QUESTION #7471

Question 1

If the vectors \(\vec{AB} = 3\hat{i} + 4\hat{k}\) and \(\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}\) are the sides of a triangle ABC, then the length of the median through A is:

\(\sqrt{18}\)
\(\sqrt{72}\)
\(\sqrt{33}\)✔️
\(\sqrt{45}\)

Correct Answer Explanation

The median from A goes to midpoint M of BC. Position vector of B from A is \(\vec{AB} = 3\hat{i}+4\hat{k}\), and of C from A is \(\vec{AC} = 5\hat{i}-2\hat{j}+4\hat{k}\). Midpoint M has position vector \(\frac{\vec{AB}+\vec{AC}}{2} = \frac{(3\hat{i}+4\hat{k})+(5\hat{i}-2\hat{j}+4\hat{k})}{2} = \frac{8\hat{i}-2\hat{j}+8\hat{k}}{2} = 4\hat{i}-\hat{j}+4\hat{k}\). Length of median \(AM = |\vec{AM}| = \sqrt{16+1+16} = \sqrt{33}\).

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