If the vertices of $\Delta ABC$ are $A(1, 4)$,$B(-3, 2)$,and $C(5, 0)$,then the length of the median $\overline{AD}$ is.........

To conduct Sports Day activities,in your rectangular shaped school ground $ABCD,$ lines have been drawn with chalk powder at a distance of $1 \,m$ each. $100$ flower pots have been placed at a distance of $1 \,m$ from each other along $AD,$ as shown in the figure. Niharika runs $\frac{1}{4}$ th the distance $AD$ on the $2^{nd}$ line and posts a green flag. Preet runs $\frac{1}{5}$ th the distance $AD$ on the $8^{th}$ line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags,where should she post her flag?

Find the centre of a circle passing through the points $(6, -6), (3, -7)$ and $(3, 3)$.

Find a relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the point $(3, 6)$ and $(-3, 4)$.

$AOBC$ is a rectangle whose three vertices are $A (0, 3)$,$O (0, 0)$,and $B (5, 0)$. The length of its diagonal is

If $P(x, y)$ is equidistant from the points $A(a+b, b-a)$ and $B(a-b, a+b),$ then prove that $bx = ay.$