Question

If a triangle `A B C ,A-=(1,10),` circumcenter `-=(-1/3,2/3),` and orthocentre `-=((11)/4,4/3)` , then the coordinates of the midpoint of the side opposite to `A` are `(1,-(11)/3)` (b) `(1,5)` `(1,-3)` (d) `(1,6)`
A. `(1,-11//3)`
B. `(1//5)`
C. `(1,-3)`
D. `(1,6)`

Answer 1

Correct Answer - A

Circumcenter `O-=(-1//3,2//3)` and orthocenter `H-=(11//3,4//3)`

Therefore the coordinates of G are `(1,8//9)`. Now,the point A is `(1,10)` as G is `(1,8//9)`.
Also, `AD:DG=3:1`,
`therefore D-=((3-1)/(3-1),(8//3-10)/(3-1))-=(1,-11//3)`
`therefore D=(3-1)/(2)=1,D_(y)=(8//3-10)/(2)=-11/3`
Hence, the coordinates of the midpoint of BC are `(1,-11//3)` .