G is the centroid of a triangle ABC, whose sides AB = 35 cm, BC = 12 cm, and AC = 37 cm. The length of BG is (correct to one decimal place):

Solution (By Examveda Team)

AB = 35 cm, BC = 12 cm and AC = 37 cm
12, 35 and 37 are triplet

ABC is a right angle triangle at ∠B = 90°
Use appoloneous theorem
$$\eqalign{ & A{B^2} + B{C^2} = 2\left( {A{D^2} + B{D^2}} \right) \cr & {35^2} + {12^2} = 2 \times {\left( {\frac{{AC}}{2}} \right)^2} + 2B{D^2} \cr & {37^2} = 2 \times {\left( {\frac{{37}}{2}} \right)^2} + 2B{D^2} \cr & 2B{D^2} = {37^2}\left( {1 - \frac{1}{2}} \right) \cr & 2B{D^2} = 1369 \times \frac{1}{2} \cr & B{D^2} = \frac{{1369}}{4} \cr & BD = \frac{{37}}{2} \cr & BG = BD \times \frac{2}{3} \cr & BG = \frac{{37}}{2} \times \frac{2}{3} \cr & BG = \frac{{37}}{3} = 12.3{\text{ cm}} \cr} $$