The point equidistant from the vertices of triangle is called

In the figure, $ABCD$ is a unit square. A circle is drawn with centre $O$ on the extended line $CD$ and passing through $A.$ If the diagonal $AC$ is tangent to the circle, then the area of the shaded region is

Construct a circumcircle of $△ABC$, where $AB=8 cm, BC=5 cm$and $\angle ABC = 60°$degree. Write the steps of construction.

Let $A_1, A_2$ and $A_3$ be the regions on $R^2$ defined by

$A_1=\left\{\left(x,y\right):x\ge 0,y\ge 0,2x+2y-x^2-y^2>1>x+y\right\}$

$A_2=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y>1>x^2+y^2\right\}$

$A_3=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y>1>x^3+y^3\right\}$

Denote by $\left|A_1\right|, \left|A_2\right|$ and $\left|A_3\right|$ the areas of the regions $A_1, A_2$ and $A_3$ respectively. Then

Let $AB$ be a sector of a circle with centre $O$ and radius $d.\angle AOB=\theta \left(<\frac{\pi }{2}\right)$ and $D$ be a point on $OA$ such that $BD$ is perpendicular$OA$. Let $E$ be the mid-point of $BD$ and $F$ be a point on the arc $AB$ such that $EF$ is parallel to $OA$. Then, the ratio of length of the arc$AF$ to the length of the arc $AB$ is

A semi - circle of diameter $1$ unit sits at the top of a semi-circle of diameter $2$ units. The shaded region inside the smaller semi-circle but outside the larger semi-circle is called a lune . The area of the lune is -

The medians of a triangle are _____.

The median of a triangle is the line segment joining any of its vertices to the _____ of the opposite side.