Find the distance between the following pairs of points: $(-5, 7)$ and $(-1, 3)$. (in $\sqrt{2}$)

Let $A(4, 2)$,$B(6, 5)$,and $C(1, 4)$ be the vertices of $\Delta ABC$. What do you observe about the point $P$ which divides the median $AD$ in the ratio $2:1$?

Find the area of the quadrilateral whose vertices,taken in order,are $(-4,-2), (-3,-5), (3,-2),$ and $(2,3)$ (in square units).

In a classroom,$4$ friends are seated at the points $A, B, C$ and $D$ as shown in the figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli,"Don't you think $ABCD$ is a square?" Chameli disagrees. Using the distance formula,find which of them is correct.

The vertices of a $\Delta ABC$ are $A(4, 6)$,$B(1, 5)$,and $C(7, 2)$. $A$ line is drawn to intersect sides $AB$ and $AC$ at $D$ and $E$ respectively,such that $\frac{AD}{AB} = \frac{AE}{AC} = \frac{1}{4}$. Calculate the area of $\Delta ADE$ and compare it with the area of $\Delta ABC$.

Show that the points $(1,7), (4,2), (-1,-1)$ and $(-4,4)$ are the vertices of a square.