If the equation of the base of an equilateral triangle is $x+y=6$ and the opposite vertex is the point $(-1, -1)$,then the area of the triangle is equal to $k$ sq. units. Find the value of $k$.

Let the mirror image of point $A(\alpha, \beta)$ in the line mirror $x + 2y = 3$ be point $B$,and the image of $B$ in the line $3x - 2y = 5$ be $C$. If the origin is the orthocentre of triangle $ABC$ and $P(a, b)$ is a point inside the triangle such that triangles $PAB$,$PBC$,and $PCA$ have the same area,then $3(a + b)$ is:

In the given figure,$AB = 12 \, cm$,$CD = 8 \, cm$,$BD = 20 \, cm$,and $\angle ABD = \angle AEC = \angle EDC = 90^{\circ}$. If $BE = x$,then:

$A$ regular octagon is formed by cutting congruent isosceles right-angled triangles from the corners of a square. If the square has side length $1$,the side length of the octagon is

Consider four triangles with side lengths $(5, 12, 9)$,$(5, 12, 11)$,$(5, 12, 13)$,and $(5, 12, 15)$. Among these,the triangle with the maximum area has sides:

Let the midpoints of the sides of a triangle $ABC$ be $(\frac{5}{2}, 7)$, $(\frac{5}{2}, 3)$, and $(4, 5)$. If its incentre is $(h, k)$, then $3h + k$ is equal to: