2d geometry system practice · Easy

If $\Delta_1$ is the area of the triangle formed by the centroid and two vertices of a triangle, $\Delta_2$ is the area of the triangle formed by the mid-points of the sides of the same triangle, then $\Delta_1: \Delta_2=$

If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle, is

The area of the quadrilateral whose vertices are $(1,2),(6,2),(5,3)$ and $(3,4)$, is

The locus of a point which moves in such a way that its distance from $(0,0)$ is three times its distance from the $x$ - axis, as given by

The triangle formed by $P(-2,3), Q(3,-2)$ and $R(0,0)$ is

The area of the pentagon, whose vertices are $A(1,1), B(7,21), C(7,-3), D(12,2)$ and $E(0,-3)$, is

Three distinct points $A, B$ and $C$ are given in the 2-dimensional coordinates plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $\frac{1}{3}$. Then, the circumcentre of the triangle $A B C$ is at the point

The centroid of a triangle lies at the origin and the coordinates of its two vertices are $(-8,7)$ and $(9,4)$. The area of the triangle is