In the following figure, a rectangle ABCD is inscribed in a circle of radius r. Given $\angle \mathrm{DAE}=30°$ and $\angle \mathrm{ACD}=30°$

What is the area of $∆AEC$ ?

In the given figure, $\angle QRN=40°, \angle PQR=46°$ and $MN$ is a tangent at $R$. What is the value(in degrees) of $x, y$ and $z$ respectively?

From a point $P$ outside a circle, $PAB$ is a secant and $PT$ is a tangent to the circle, where $A, B$ and $T$ are points on the circle. If $PT=5 cm, PA=4 cm$ and $AB=x cm$, then $x$ is equal to :

Consider the following diagram in which $AB$ is the tangent to circle. Find the length of $OQ(in cm)$ if the measure of the largest chord of that circle is $6 cm$ and $PQ$ is $4 cm.$

What is the circumference of the largest circle which can be inscribed in a square of side $14 cm$? ($\mathrm{\pi }=\frac{22}{7}$)

The two circles with radii $6 cm$ and $3 cm$ centred at the point $P$ and $Q$ respectively. A transverse common tangent $AC$ to the both circles cuts line Joining $P$ and $Q$ at the point $B$ such that $AB = 8 cm$. Calculate the length (in cm.) of $PQ$.

In the given figure, AP and BP are tangents to a circle with centre O. If $\angle APB={62}^{\circ }$ then the measure of $\angle AQB$ is:

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