If O, G, S are respectively the orthocentre, centroid and circumcentre of a triangle whose vertices are A(2,3), B(2,4) and C(4,3) , then A O^2+9 B G^2+4 C S^2=

Step-by-step solution

Coordinate of vertices of triangle aligned & A(2,3), B(2,4), C(4,3) & A B=1, B C= 5 , C A=2 aligned So, A B C is right angle triangle where right angle at A that is orthocentre also. Coordinate of orthocentre is O(2,3) . Coordinates of centroid aligned & = ( x₁+x₂+x₃ 3 , y₁+y₂+y₃ 3 ) G & = ( 8 3 , 10 3 ) aligned G divide the line joining O and S in the ratio 2: 1 gathered 8 3 = 2 x+2 3 x=3 10 3 = 2 y+3 3 y= 7 2 S (3, 7 2 ) So, A O^2+9 B G^2+4 C S^2 A O^2=(2-2)^2+(3-3)^2=0 A O^2=0 B G^2= (2- 8 3 )^2+ (4- 10 3 )^2= 8