The vertices of $\Delta ABC$ are $A(a, 6)$, $B(5, 1)$, and $C(4, 6)$. If the circumcentre of the triangle is $P(2, 3)$, find the value of $a$.

Ayush starts walking from his house to his office. Instead of going to the office directly,he goes to a bank first,from there to his daughter's school,and then reaches the office. What is the extra distance travelled by Ayush in reaching his office (in $, km$)? (Assume that all distances covered are in straight lines).
If the house is situated at $(2,4)$,the bank at $(5,8)$,the school at $(13,14)$,and the office at $(13,26)$,where coordinates are in $km$.

State whether the following statement is true or false. Justify your answer.
The point $A (2,7)$ lies on the perpendicular bisector of the line segment joining the points $P (6,5)$ and $Q (0,-4)$.

Name the type of quadrilateral formed,if any,by the following points,and give reasons for your answer: $(-1,-2), (1,0), (-1,2), (-3,0)$.

Show that $(2,2), (5,2), (5,5)$ and $(2,5)$ are the vertices of a square.

If the vertices of $\square ABCD$ are $A(1, 3), B(4, 3), C(4, 5)$ and $D(1, 5)$,then $\square ABCD$ is a $\ldots$