AP EAMCET · Maths · Vector Algebra

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of the vertices of a triangle $A B C$ . Through the vertices, lines are drawn parallel to the sides to form the triangle $A^{'} B^{'} C^{'}$ . Then the centroid of $Δ A^{'} B^{'} C^{'}$ is

A $\frac{\vec{a} + \vec{b} + \vec{c}}{9}$
B $\frac{\vec{a} + \vec{b} + \vec{c}}{6}$
C $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$
D $\frac{2 (\vec{a} + \vec{b} + \vec{c})}{3}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

Step-by-step Solution

Detailed explanation

Here, A is mid-point of B'C', B is mid-point of A'C' and C is mid-point of A'B'. By mid-point formula a→=b'→+c'→2, b→=a'→+c'→2, c→=b'→+a'→2 Centroid of triangle ABC,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let a→, b→, c→ be the position vectors of the vertices of a triangle ABC. Through the vertices, lines are drawn parallel to the sides to form the triangle A'B'C'. Then the centroid of ΔA'B'C' is

Step-by-step Solution

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of the vertices of a triangle $A B C$ . Through the vertices, lines are drawn parallel to the sides to form the triangle $A^{'} B^{'} C^{'}$ . Then the centroid of $Δ A^{'} B^{'} C^{'}$ is