The diagonal passing through the origin of a quadrilateral formed by $x = 0, y = 0, x + y = 1$ and $6x + y = 3$ is

Let $ABC$ be a triangle with $\angle B = 90^{\circ}$. Let $AD$ be the bisector of $\angle A$ with $D$ on $BC$. Suppose $AC = 6 \text{ cm}$ and the area of the $\triangle ADC$ is $10 \text{ cm}^2$. Then,the length of $BD$ in $\text{cm}$ is equal to

The equations of two altitudes of an equilateral triangle are $\sqrt{3}x - y + 8 - 4\sqrt{3} = 0$ and $\sqrt{3}x + y - 12 - 4\sqrt{3} = 0$. The equation of the third altitude is

In a quadrilateral $ABCD$,which is not a trapezium,it is known that $\angle DAB = \angle ABC = 60^{\circ}$. Moreover,$\angle CAB = \angle CBD$. Then,

In an isosceles right-angled triangle,if the equation of the hypotenuse is $3x + 4y = 4$ and its opposite vertex is $(2, 2)$,then the slopes of the remaining two sides are:

The point $P(3,6)$ is first reflected on the line $y=x$ and then the image point $Q$ is again reflected on the line $y=-x$ to get the image point $Q^{\prime}$. Then,the circumcentre of the $\Delta P Q Q^{\prime}$ is