question_answer

ABC is an isosceles triangle with AB = AC = 5 and BC = 6. If G is the centroid of \[\Delta \,ABC\], then AG is equal to

A) \[\frac{1}{3}\]

B) \[\frac{2}{3}\]

C) \[\frac{4}{3}\]

D) \[\frac{8}{3}\]

Correct Answer: D

Solution :

 Let      \[GD=x\]  so that      \[AG=2x\] \[\therefore \]  \[AD=x+2x=3x\] Since AD is a median, therefore                 \[BD=\frac{1}{2}\times BC=\frac{1}{2}\times 6=3\] From right angled \[\Delta ABC\] \[A{{B}^{2}}=B{{D}^{2}}+A{{D}^{2}}\] or            \[{{5}^{2}}={{3}^{2}}+{{(3x)}^{2}}\] or            \[25=9+9{{x}^{2}}\] or           \[9{{x}^{2}}=16\] or            \[{{x}^{2}}=\frac{16}{9}\] \[\therefore \]  \[x=\frac{4}{3}\]                 Hence,      \[AG=2x\]                 \[=2\times \frac{4}{3}=\frac{8}{3}\]