Question

21. If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is
A. `(1,(7)/(3))`
B. `((1)/(3),2)`
C. `((1)/(3),1)`
D. `((1)/(3),(5)/(3))`

Answer 1

Let a `DeltaABC` is such that vertices ltbr. `A(1,2)`, `B(x_(1),y_(1))` and `C(x_(2),y_(2))`.

It is given that mid-point of side `AB` is `(-1,1)`.
So, `(x_(1)+1)/(2)=-1`
and `(y_(1)+2)/(2)=1`
`impliesx_(1)=-3` and `y_(1)=0`
So, point `B` is `(-3,0)`
Also, it is given that mid-point of side `AC` is `(2,3)`, so
`(x_(2)+1)/(2)=2` and `(y_(2)+2)/(2)=3`
`impliesx_(2)=3` and `y_(2)=4`
So, point `C` is `(3,4)`.
Now, centroid of `DeltaABC` is
`G((1+(-3)+3)/(3),(2+0+4)/(3))=G((1)/(3),2)`