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 #7435

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 the midpoint M of BC. We have \(\vec{AB} + \vec{BC} = \vec{AC}\), so \(\vec{BC} = \vec{AC} - \vec{AB} = (5\hat{i}-2\hat{j}+4\hat{k}) - (3\hat{i}+4\hat{k}) = 2\hat{i}-2\hat{j}\). The midpoint M of BC from B is at \(\frac{\vec{BC}}{2} = \hat{i}-\hat{j}\). The position of M from A is \(\vec{AM} = \vec{AB} + \frac{\vec{BC}}{2} = 3\hat{i}+4\hat{k} + \hat{i}-\hat{j} = 4\hat{i}-\hat{j}+4\hat{k}\). The length is \(|\vec{AM}| = \sqrt{16+1+16} = \sqrt{33}\).

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