Shown below is a circle with multiple chords. One of the chords is the diameter of the circle.

Find the measure of the angle subtended by a $4.7\pi \mathrm{cm}$ arc at the circumference of the circle. Show your work and give valid reasons.

In the figure given below $O$ is the centre of the circle. If the $QR=OP$ and angle $\angle ORP=20°$. Find the value of $x$ giving reasons.

In a circle of radius $21 \mathrm{cm}$, an arc subtends an angle of $60°$ at the centre. Find the length of the arc.

Prove that, the angles in the same segment of a circle are equal to each other.

In the given figure, $ABCDE$ is a pentagon inscribed in a circle such that $AC$ is a diameter and side $BC\parallel AE$. If $\angle BAC=50°$, find giving reason

$\angle BEC$

Prove that the angle formed by an arc on the centre of the circle is twice the angle formed at any point of remaining part of the circle.

In a circle of radius $21$cm an arc subtends an angle of $60°$ at the centre. Find the length of the arc.

In the given figure $TP$ and $TQ$ are two tangents to the circle with centre $O$, touching at $A$ and $C$ respectively. If $\angle BCQ=55°$ and $\angle BAP=60°$, find$\angle AOC.$

In the given figure a quadrilateral $ABCD$ have $AB=AC=AD$, then prove that $\angle BAD=2(\angle BDC+\angle CBD)$.

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