Find the ratio in which the $y$-axis divides the line segment joining the points $(5, -6)$ and $(-1, -4)$. Also,find the point of intersection.

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

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

The Class $X$ students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of $1\, m$ from each other. There is a triangular grassy lawn in the plot as shown in the Figure. The students are to sow seeds of flowering plants on the remaining area of the plot. Taking $A$ as origin,find the coordinates of the vertices of the triangle.

$A$ median of a triangle divides it into two triangles of equal areas. Verify this result for $\Delta ABC$ whose vertices are $A (4, -6), B (3, -2)$ and $C (5, 2)$. Find the area of each triangle (in square units).

If $A$ and $B$ are $(-2, -2)$ and $(2, -4)$ respectively,find the coordinates of $P$ such that $AP = \frac{3}{7} AB$ and $P$ lies on the line segment $AB$.