Determine the ratio in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2)$ and $B(3, 7)$.

If $(1, 2), (4, y), (x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order,find $x$ and $y$.

Find the distance between the following pairs of points: $(2,3)$ and $(4,1)$.

Find a relation between $x$ and $y$ if the points $(x, y), (1, 2)$ and $(7, 0)$ are collinear.

Name the type of quadrilateral formed,if any,by the following points,and give reasons for your answer: $(4,5), (7,6), (4,3), (1,2)$.

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.