Question

If the middle points of sides BC, CA and AB of triangle ABC are respectively D, E, F. If the position vectors of A, B, C are \(\hat{i}+\hat{j},\;\hat{j}+\hat{k},\;\hat{k}+\hat{i}\) respectively, then the position vector of the centre of triangle DEF is

• A. \(\dfrac{1}{3}(\hat{i}+\hat{j}+\hat{k})\)
• B. \(\hat{i}+\hat{j}+\hat{k}\)
• C. \(2(\hat{i}+\hat{j}+\hat{k})\)
• D. \(\dfrac{2}{3}(\hat{i}+\hat{j}+\hat{k})\)

Hint:

The centroid of the medial triangle is same as the original triangle.

Answer 1

The Correct Option is A

Step 1: Coordinates of centroid of ABC: \[ \frac{A+B+C}{3} \]
Step 2: Triangle DEF has same centroid as ABC.
Step 3: \[ \frac{(\hat{i}+\hat{j})+(\hat{j}+\hat{k})+(\hat{k}+\hat{i})}{3} =\frac{1}{3}(\hat{i}+\hat{j}+\hat{k}) \]