The equation of one side of an equilateral triangle is $x+y=2$ and one vertex is $(2,-1)$. The length of the side is

Let $P$ be an interior point of a convex quadrilateral $ABCD$ and $K, L, M, N$ be the mid-points of $AB, BC, CD, DA$ respectively. If $\text{Area}(PKAN) = 25$,$\text{Area}(PLBK) = 36$,and $\text{Area}(PMDN) = 41$,then $\text{Area}(PLCM)$ is

Let $A(\alpha, -2)$,$B(\alpha, 6)$,and $C\left(\frac{\alpha}{4}, -2\right)$ be the vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$,then which of the following is $NOT$ correct about $\triangle ABC$?

The coordinates of the vertices $P, Q, R$ and $S$ of a square $PQRS$ inscribed in the triangle $ABC$ with vertices $A \equiv (0, 0)$,$B \equiv (3, 0)$,and $C \equiv (2, 1)$,given that two of its vertices $P$ and $Q$ lie on the side $AB$,are respectively:

If the sides $AB, BC, CD$ and $DA$ of a quadrilateral are given by the equations $x + 2y = 3, x = 1, x - 3y = 4$ and $5x + y + 12 = 0$ respectively,then find the angle between the diagonals $AC$ and $BD$ in degrees.

Let $ABC$ be a triangle and $M$ be a point on side $AC$ closer to vertex $C$ than $A$. Let $N$ be a point on side $AB$ such that $MN$ is parallel to $BC$ and let $P$ be a point on side $BC$ such that $MP$ is parallel to $AB$. If the area of the quadrilateral $BNMP$ is equal to $\frac{5}{18}$ of the area of $\triangle ABC$,then the ratio $AM/MC$ equals