Circles

In the diagram, radii $OA$ and $OB$ meet at the center $O$. What is the term for the 'pie-shaped' region bounded by these two radii and the arc $AB$?

The region bounded by a chord and either of its arcs is called a segment. In the diagram, the shaded region between chord $XY$ and the smaller arc is known as the:

A line segment $PQ$ connects two points on the boundary of the circle but does not pass through the center. What is this line segment called?

In the given circle with center $O$, if $AB$ is a line segment passing through $O$ and ending at the boundary of the circle, what is the specific term for $AB$?

If a circle is divided into three equal sectors by three radii, what is the central angle of each sector?

A circle is divided into two parts by a diameter. If the length of the diameter is $14$ cm, what is the length of the boundary of one of the resulting semicircles (including the diameter edge)? Take $\pi = \frac{22}{7}$.

Given two concentric circles, the region between them is known as an:

Three circles of equal radius $r$ are placed such that each circle touches the other two externally. What is the area of the region enclosed between the three circles?

A square of side $14$ cm is inscribed in a circle. Four semi-circles are then drawn on the four sides of the square as diameters, outward. Find the total area of the four shaded regions (the crescent shapes called lunes).

In a circle, chords $AB$ and $CD$ intersect at right angles at point $E$ inside the circle. If $AE = 2$ cm, $EB = 6$ cm, and $ED = 3$ cm, find the diameter of the circle.

In a circle with center $O$, $OM \perp AB$. If $AM = 3.5\text{ cm}$, what is the length of chord $AB$?

If two chords of a circle are equal, then their corresponding arcs are:

A chord of a circle is equal to its radius. What is the angle subtended by this chord at the center?

In a circle with center $O$, chord $AB$ is $8\text{ cm}$ and its distance from the center is $3\text{ cm}$. What is the radius of the circle?

In the given figure, $O$ is the center of a circle. A chord $AB$ of length $16$ cm is at a distance of $6$ cm from the center $O$. A perpendicular $OM$ is drawn from $O$ to $AB$. What is the radius of the circle?

Two equal chords $AB$ and $CD$ of a circle with center $O$ when produced meet at a point $P$ outside the circle. If $PB = 3\text{ cm}$ and $AB = 5\text{ cm}$, what is the length of $PD$ (given $D$ is closer to $P$ than $C$)?

Two chords $AB$ and $CD$ are such that $AB = 10\text{ cm}$ and $CD = 24\text{ cm}$. If the radius of the circle is $13\text{ cm}$, find the distance of the longer chord from the center.

A park has a circular walking track. A straight fence $PQ$ of length $16$ m is installed inside the park as a shortcut between two points on the track. A security camera is mounted at the center $O$ of the circular track. The perpendicular distance from the camera $O$ to the fence $PQ$ is $6$ m. If another straight fence $RS$ of length $12$ m is to be installed in the same park, what will be its perpendicular distance from the camera $O$?

A circular arc of a railway track has a radius of $1500$ m. If a train travels a distance of $440$ m along the track, what is the angle (in degrees) subtended by this arc at the center? (Take $\pi = 22/7$)

Two circles of radii $5$ cm and $3$ cm intersect at two points and the distance between their centers is $4$ cm. Find the length of the common chord.

Two chords of a circle are at a distance of $2$ cm from the center. If one chord is $6$ cm, the other chord is:

If the distance of a chord from the center of a circle increases, then the length of the chord:

In a circle, if chord $AB = 12$ cm and its distance from center $O$ is $8$ cm, find the radius of the circle.

The longest chord of a circle is called its:

In a circle of radius $5$ cm, $AB$ and $CD$ are two parallel chords of length $8$ cm and $6$ cm respectively. If they are on opposite sides of the center, the distance between them is:

Three points $A, B$ and $C$ are on a circle. If $AB = AC = 6$ cm and radius is $5$ cm, the length of chord $BC$ is:

If two chords of a circle are equally inclined to the diameter through their point of intersection, then the chords are:

A circular track has two straight segments (chords) $XY$ and $ZW$ that are equal in length. If the perpendicular distance from the center of the track to $XY$ is $(3x - 5)$ m and to $ZW$ is $(x + 7)$ m, find the actual distance of these chords from the center.

In a circle with center $O$, chord $AB$ and $CD$ are such that $AB = 10$ cm and $CD = 24$ cm. If the radius is 13 cm and $AB \parallel CD$, what is the distance between the chords if they are on opposite sides of the center?

Two parallel chords of a circle of length 30 cm and 16 cm are on opposite sides of the center. If the distance between them is 23 cm, find the radius of the circle.

In the given figure, $O$ is the center of the circle. If the central angle $\angle AOB = 80^\circ$, find the measure of the angle $\angle ACB$ subtended by the same arc at a point $C$ on the remaining part of the circle.

In a circle with center $O$, $\angle AOB = 60^\circ$. What is the measure of $\angle APB$ where $P$ is a point on the major arc?

Which of the following describes the relationship between the angle subtended by an arc at the center and at the circumference?

In the given figure, $O$ is the center of the circle. If $\angle PQR = 130^\circ$, find the reflex $\angle POR$.

If $\angle CAD = 40^\circ$ and $\angle ACB = 70^\circ$ in a cyclic quadrilateral $ABCD$, find $\angle ABC$ if $AD \parallel BC$.

If $\angle AOB = 90^\circ$ and $\angle BOC = 120^\circ$ in a circle with center $O$, find $\angle ABC$.

A cyclic quadrilateral $ABCD$ is such that $AB$ is the diameter of the circle. If $\angle ADC = 120^\circ$, find $\angle BAC$.

A circular shield has an inscribed triangle $PQR$. If $PQ$ is the diameter of the circle and $R$ is any point on the semi-circle, and it is given that $\angle RPQ = 30^\circ$, calculate the measure of $\angle RQP$.

In a circle with center $O$, the chords $AB$ and $CD$ are equal. If $\angle AOB = 80^\circ$, find the measure of the angle subtended by chord $CD$ at the circumference in the major segment.

A circular track has points $A$, $B$, $C$, $D$ in cyclic order. If the diagonal $AC$ is a diameter and $\angle BAC = 35^\circ$, calculate $\angle CAD$ given that $ABCD$ is a rectangle.

Points $A, B, C, D$ are on a circle. If $AC$ and $BD$ are diameters, what shape is $ABCD$?

In the figure, $\angle ABC = 120^{\circ}$, where $A, B, C, D$ are points on the circle. Find $\angle ADC$.

If $ABCD$ is a cyclic parallelogram, then it must be a:

If an angle of a cyclic quadrilateral is $90^{\circ}$, then the side opposite to it must be a:

In the given figure, $ABCD$ is a cyclic quadrilateral where diagonals $AC$ and $BD$ intersect at point $E$. If $\angle DBC = 65^\circ$ and $\angle BAC = 45^\circ$, find the measure of $\angle BCD$.

If two opposite angles of a cyclic quadrilateral are $x + 10$ and $2x - 40$, find $x$.

Given a circle with center $O$. $A, B, C$ are points on the circle. If $\angle ABC = 100^{\circ}$, find the reflex $\angle AOC$.

In a circle with center $O$, chord $AB$ is equal to the radius of the circle. A point $C$ is on the major arc $AB$. If $ABCD$ is a cyclic quadrilateral where $D$ is a point on the minor arc $AB$, find the measure of $\angle ADB$.

A cyclic trapezium $ABCD$ has $AB \parallel CD$. if $\angle A = (2x - 10)^\circ$ and $\angle C = (x + 40)^\circ$, determine the value of $x$.

Two circles intersect at points $M$ and $N$. From $M$, two line segments $PMQ$ and $RMS$ are drawn such that $P, R$ lie on one circle and $Q, S$ lie on the other. If $\angle PRM = 85^\circ$, find $\angle MSN$.