Question

If $G$ and $G^{\prime }$ be the centroids of the triangles $ABC$ and $A^{\prime }{B}^{\prime }{C}^{\prime }$ respectively, then $\overrightarrow{A{A}^{\prime }}+\overrightarrow{B{B}^{\prime }}+\overrightarrow{C{C}^{\prime }}=$

• A. $\frac{2}{3}\overrightarrow{G{G}^{\prime }}$
• B. $\overrightarrow{G{G}^{\prime }}$
• C. $2\overrightarrow{G{G}^{\prime }}$
• D. $3\overrightarrow{G{G}^{\prime }}$

Answer 1

The correct option is D $3\overrightarrow{G{G}^{\prime }}$

Let P.V. of $A\left(\overrightarrow{a}\right),B\left(\overrightarrow{b}\right),C\left(\overrightarrow{c}\right),A^{\prime }\left(\overrightarrow{a_1}\right),B^{\prime }\left(\overrightarrow{b_1}\right),C^{\prime }\left(\overrightarrow{c_1}\right)$

Now P.V. of $G=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}$;

Similarly $G^{\prime }=\frac{\overrightarrow{a_1}+\overrightarrow{b_1}+\overrightarrow{c_1}}{3}$

Now $\overrightarrow{A{A}^{\prime }}=\overrightarrow{a_1}-\overrightarrow{a},\overrightarrow{B{B}^{\prime }}=\overrightarrow{b_1}-\overrightarrow{b},\overrightarrow{C{C}^{\prime }}=\overrightarrow{c_1}-\overrightarrow{c}$

So $\overrightarrow{A{A}^{\prime }}+\overrightarrow{B{B}^{\prime }}+\overrightarrow{C{C}^{\prime }}=(\overrightarrow{a_1}+\overrightarrow{b_1}+\overrightarrow{c_1})-(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$

$=3(\frac{\overrightarrow{a_1}+\overrightarrow{b_1}+\overrightarrow{c_1}}{3}-\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3})=3\overrightarrow{G{G}^{\prime }}$