Solveeit Logo
Login

Question

Question: If G and G' be the centroids of the triangles ABC and \(A'B'C'\) respectively, then \(\overset{\righ...

If G and G' be the centroids of the triangles ABC and A′B′C′A'B'C' respectively, then AA→′+BB′→+CC→′=\overset{\rightarrow}{AA}' + \overset{\rightarrow}{BB'} + \overset{\rightarrow}{CC}' =

A

23GG→′\frac{2}{3}\overset{\rightarrow}{GG}'

B

GG→′\overset{\rightarrow}{GG}'

C

2GG→′2\overset{\rightarrow}{GG}'

D

3GG→′3\overset{\rightarrow}{GG}'

Answer

3GG→′3\overset{\rightarrow}{GG}'

Explanation

Solution

GA→+GB→+GC→=0\overset{\rightarrow}{GA} + \overset{\rightarrow}{GB} + \overset{\rightarrow}{GC} = \mathbf{0} and G′A′→+G′B′→+G′C′→=0\overset{\rightarrow}{G^{'}A^{'}} + \overset{\rightarrow}{G^{'}B^{'}} + \overset{\rightarrow}{G^{'}C^{'}} = \mathbf{0}

⇒(GA→−G′A′→)+(GB→−G′B′→)+(GC→−G′C′→)=0\Rightarrow (\overset{\rightarrow}{GA} - \overset{\rightarrow}{G^{'}A^{'}}) + (\overset{\rightarrow}{GB} - \overset{\rightarrow}{G^{'}B^{'}}) + (\overset{\rightarrow}{GC} - \overset{\rightarrow}{G^{'}C^{'}}) = \mathbf{0}

⇒(GA→+G′G→−G′A′→)+(GB→+G′G→−G′B′→)+(GC→+G′G→−G′C′→)=3G′G→\Rightarrow (\overset{\rightarrow}{GA} + \overset{\rightarrow}{G^{'}G} - \overset{\rightarrow}{G^{'}A^{'}}) + (\overset{\rightarrow}{GB} + \overset{\rightarrow}{G^{'}G} - \overset{\rightarrow}{G^{'}B^{'}}) + (\overset{\rightarrow}{GC} + \overset{\rightarrow}{G^{'}G} - \overset{\rightarrow}{G^{'}C^{'}}) = 3\overset{\rightarrow}{G^{'}G}

⇒(GA→−GA′→)+(GB→−GB′→)+(GC→−GC′→)=3G′G→\Rightarrow (\overset{\rightarrow}{GA} - \overset{\rightarrow}{GA^{'}}) + (\overset{\rightarrow}{GB} - \overset{\rightarrow}{GB^{'}}) + (\overset{\rightarrow}{GC} - \overset{\rightarrow}{GC^{'}}) = 3\overset{\rightarrow}{G^{'}G}

⇒A′A→+B′B→+C′C→=3G′G→⇒AA′→+BB′→+CC′→=3GG′→\Rightarrow \overset{\rightarrow}{A^{'}A} + \overset{\rightarrow}{B^{'}B} + \overset{\rightarrow}{C^{'}C} = 3\overset{\rightarrow}{G^{'}G} \Rightarrow \overset{\rightarrow}{AA^{'}} + \overset{\rightarrow}{BB^{'}} + \overset{\rightarrow}{CC^{'}} = 3\overset{\rightarrow}{GG^{'}}.