Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$-axis. Also,find the coordinates of the point of division.

Check whether $(5,-2), (6,4),$ and $(7,-2)$ are the vertices of an isosceles triangle.

Find the area of a triangle formed by the points $A(5, 2)$,$B(4, 7)$,and $C(7, -4)$ (in square units).

In what ratio does the point $(-4, 6)$ divide the line segment joining the points $A(-6, 10)$ and $B(3, -8)$?

Find the coordinates of a point $A,$ where $AB$ is the diameter of a circle whose centre is $(2,-3)$ and $B$ is $(1,4).$

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?