The question involves computing or locating special points of a triangle (centroid, orthocenter, incenter, circumcenter) or using properties like area, angle bisectors, or perpendicular bisectors within a triangle defined by coordinates or lines.
In the rectangular coordinate plane, one vertex of a triangle is at the origin, its centroid is at the point $( 0,6 )$, and its orthocenter is at the point $( 0,8 )$. Accordingly, what is the area of this triangle in square units? A) 18 B) 21 C) 24 D) 27 E) 30
In a rectangular coordinate plane, points $A(9,2)$, $B(10,1)$, $C$, $D(4,13)$, $E(3,6)$ and $F$ are given. Given that the centroid of triangle $ABC$ and the centroid of triangle $DEF$ are the same point, what is the distance between points $C$ and $F$ in units? A) 10 B) 13 C) 15 D) 17 E) 20