The area of a triangle having vertices $A(3, 0)$,$B(0, 3)$,and $C(3, 3)$ is:

If $G$ is the centroid of a triangle having vertices $A(3h, 3k)$,$B(-3a, 0)$,and $C(3a, 0)$,then prove that $AB^2 + BC^2 + AC^2 = 3(GA^2 + GB^2 + GC^2)$.

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 $km$.

$P(3, 2)$ and $Q(7, k)$ are the given points. If $d(P, Q) = 5$,then $k = \dots \dots \dots$

Find the circumcentre of the triangle with vertices $(4,6), (5,1),$ and $(4,0)$.

Find the area of the triangle whose vertices are $(-8, 4), (-6, 6)$ and $(-3, 9).$