I'm Up and Down, and Round and Round

A line parallel to the $y$-axis moves ________ compared to the $y$-axis.

A line parallel to the $x$-axis moves ________ compared to the $x$-axis.

If a point lies on the $y$-axis, its $x$-coordinate is always ________.

In the point $(-5, 7)$, the ordinate is ________.

The point $(-4, -4)$ is at the same distance from both axes. In which quadrant is it located?

If a line intersects two concentric circles with center $O$ at points $A, B, C,$ and $D$, then $AB$ is always equal to ____.

What is the name of the system used for locating a point in a plane using two perpendicular lines?

A point $P$ lies in the first quadrant such that its abscissa is double its ordinate. Point $P$ is reflected across the $y$-axis to obtain point $Q$, and point $Q$ is then reflected across the $x$-axis to obtain point $R$. If the area of the triangle $\triangle PQR$ is $16$ square units, what are the coordinates of point $R$?

In the Cartesian plane, if we move 'round and round' a circle centered at $(0,0)$ with radius $5$, which of the following points will we never pass through?

Which of the following points is equidistant from the axes and lies in the second quadrant?

If the distance between the center and a chord decreases, the length of the chord:

What is the angle in a major segment of a circle?

The angle subtended by a minor arc in the alternate segment is:

Which shape possesses infinite rotational symmetry?

Points $A$, $B$, and $C$ lie on a circle with center $O$. If $B$ is a point on the minor arc $AC$ and $\angle ABC = 130^\circ$, what is the measure of the non-reflex $\angle AOC$?

Two concentric circles are drawn with radii $10$ cm and $6$ cm. What is the length of a chord of the larger circle that is also tangent to the smaller circle?

If a diameter of a circle bisects a chord (which is not itself a diameter), what is the measure of the angle formed between that diameter and the chord?

In a circle of radius $17$ cm, two parallel chords are drawn on opposite sides of the center. The lengths of the chords are $16$ cm and $30$ cm. Find the distance between the chords.

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

In a circle, two chords $AB$ and $AC$ are equal. Prove that the center $O$ lies on the bisector of $\angle BAC$. What is the angle between $OA$ and $BC$ if $OA$ intersects $BC$ at $M$?

The angle subtended by an arc at the center is $140^{\circ}$. What is the angle subtended by it at any point on the remaining part of the circle?

If the sum of a pair of opposite angles of a quadrilateral is $180^{\circ}$, the quadrilateral is:

In the diagram, $AB$ and $CD$ are two equal chords of a circle with center $O$. If $OM \perp AB$ and $ON \perp CD$, then which of the following is true?

How many circles can pass through three non-collinear points?

In the figure provided, $AB$ is a diameter of the circle with center $O$. If $AC$ is a chord such that $\angle BAC = 30^\circ$ and a tangent is drawn at $C$ which intersects the extended diameter $AB$ at point $P$, find the measure of $\angle APC$.

In the following figure, chords $AB$ and $CD$ of a circle intersect at point $P$ inside the circle. If $\angle APC = 80^\circ$ and the arc $AC$ subtends an angle of $100^\circ$ at the center $O$, find the angle subtended by arc $BD$ at the center $O$.

In the given figure, $O$ is the center of the circle. If $\angle ABC = 45^\circ$, what is the measure of $\angle ADC$ if $D$ lies on the major arc $ADC$?

A round table cover has six equal designs as shown in the figure. If the radius of the cover is $28$ cm, find the cost of making the designs at the rate of Rs $0.35$ per $\text{cm}^2$. (Use $\sqrt{3} = 1.7$)

Four equal circles are described about the four corners of a square so that each touches two of the others. If a side of the square is $14$ cm, find the area of the space enclosed between the circles. (Take $\pi = \frac{22}{7}$)

A chord of a circle of radius $10$ cm subtends a right angle at the center. Find the area of the minor segment. (Take $\pi = 3.14$)

How many circles can pass through three non-collinear points?

In the figure, if $OM \perp AB$ and radius $OA = 5$ cm, $OM = 3$ cm, then the length of chord $AB$ is:

A perpendicular from the center of a circle to a chord ________ the chord.

If a chord $PQ$ subtends $80^{\circ}$ at the center, what angle does it subtend at the circumference in the minor arc?

In a circle with center $O$, a chord $AB$ subtends an angle of $70^\circ$ at the center. What is the measure of the angle subtended by the same chord at any point $C$ on the major arc of the circle?

In the figure, if $O$ is the center and $\angle APC = 120^{\circ}$, what is $\angle AOC$ (major)?

Two chords $AB$ and $CD$ of a circle are parallel and a line $L$ is the perpendicular bisector of $AB$. Then $L$:

In a circle, chords $AB$ and $AC$ are such that $AB = AC = 6$ cm. If the radius of the circle is $5$ cm, find the measure of the angle $\angle BAC$ using the property of angles subtended at the center.

A circular logo has three points $X, Y, Z$ such that $XY = YZ = ZX$. What is the angle subtended by the chord $XY$ at any point on the major arc $XY$?

In the given circular diagram, $O$ is the center. Chords $AB$ and $CD$ are parallel. If $\angle AOB = 80^{\circ}$ and $\angle COD = 60^{\circ}$, and both chords are on the same side of the center, find the measure of $\angle AOC$.

In a circle with center $O$, a chord $AB$ of length $16$ cm is drawn. A line $OM$ is drawn from the center $O$ to the midpoint $M$ of the chord $AB$. If the radius of the circle is $10$ cm, what is the length of the perpendicular distance $OM$?

In a circle, if chord $AB = 12$ cm and its distance from the center is $8$ cm, what is the length of another chord $CD$ which is also at a distance of $8$ cm from the center?

How many circles can pass through three non-collinear points?

Given a circle with center $O$ and chord $AB$. If the radius is $13$ cm and the distance of the chord from the center is $5$ cm, find the length of the chord.

A point $P$ is at a distance of $10$ cm from the center of a circle. If the length of the chord through $P$ which is perpendicular to the radius through $P$ is $16$ cm, find the radius of the circle.

Given two concentric circles with center $O$. A line $l$ intersects the circles at points $A, B, C, D$ in order. If $AD = 12$ cm and $BC = 8$ cm, find the length of $AB$.

In a circle with center $O$, $AB$ and $AC$ are two chords such that $AB = AC = 6$ cm. If the radius of the circle is $5$ cm, find the length of the chord $BC$.

A designer is creating a circular window with radius $R$. He places a decorative metal bar as a chord $XY$ of length $16$ cm. He then constructs a perpendicular bisector to this chord which intersects the circle at point $P$ (the vertex of the minor arc). If the distance from the midpoint $M$ of the chord to the point $P$ is $4$ cm, find the radius $R$ of the circular window.

A circular irrigation tank has a radius of $13$ m. Two parallel pipes are submerged as chords in the tank. The first pipe, $AB$, is $24$ m long. The second pipe, $CD$, is $10$ m long and is located on the opposite side of the tank's center $O$ relative to $AB$. A maintenance sensor is placed at the midpoint $M$ of $AB$ and another at the midpoint $N$ of $CD$. What is the shortest distance between the two sensors $M$ and $N$ along a straight line passing through the center?

A diameter of a circle is also a perpendicular bisector of a chord $AB$. If the radius of the circle is $25$ units and the length of the chord $AB$ is $48$ units, find the distance from the midpoint of the chord to the nearest point on the circumference along the diameter.

Two chords are at distances of $4$ cm and $5$ cm from the centre. Which chord is longer?

A chord is $16$ cm long. The radius of the circle is $10$ cm. How far is the chord from the centre?

If the distance of a chord from the centre is $5$ cm and the diameter of the circle is $26$ cm, find the length of the chord.

What is the length of the longest chord of a circle if its radius is $7.5$ cm?

In a circle of radius $6$ cm, a chord $PQ$ is drawn such that its perpendicular distance from the centre is equal to half the length of the chord. What is the length of chord $PQ$?

A chord is at a distance of $9$ cm from the centre of a circle of radius $15$ cm. Find the length of the chord.

Two parallel chords of length $10$ cm and $24$ cm are on the same side of the centre of a circle of radius $13$ cm. What is the distance between them?

The perpendicular distance from the center of a circle to a chord of length $2\sqrt{r^2-d^2}$ is $d$. If the radius of the circle is $r$, what is the length of the chord?

If a chord of length $48\text{ cm}$ is $7\text{ cm}$ from the center, find the distance of a chord of length $40\text{ cm}$ from the center.

A chord of length $10\text{ cm}$ is at a distance of $12\text{ cm}$ from the center. Find the length of a chord at a distance of $5\text{ cm}$ from the center.

In a circle, if the central angle subtended by an arc is $x$, and the angle subtended at the circumference is $y$, which relationship is correct?

If an arc of a circle is $1/4$ of the circumference, what is the angle subtended by the arc at the center?

What is the angle subtended by a minor arc at the center if the angle subtended by it at the circumference is $48^{\circ}$?

Find the value of $x$ in the figure if $O$ is the center of the circle and $\angle OBA = 50^{\circ}$.

In the given figure, $O$ is the center of the circle. If $\angle ABC = 45^{\circ}$, then what is the value of the angle subtended by the major arc $AC$ at the center, denoted as reflex $\angle AOC$?

Two circles intersect at two points $A$ and $B$. $AC$ and $AD$ are diameters to the two circles. What is the nature of points $C, B$ and $D$?

In the figure, $\angle BDC = 45^{\circ}$ and $\angle ABC = 55^{\circ}$. Find $\angle ACB$ if $A, B, C, D$ are points on a circle.

A clock face is a circle. Let the center be $O$. At 4:00, the hour hand is at $A$ (4) and the minute hand is at $B$ (12). A point $C$ is chosen on the major arc $AB$. Find the measure of $\angle ACB$.

Two circles intersect at points $G$ and $H$. Through $G$, two line segments $AGB$ and $CGD$ are drawn to intersect the circles at $A, B$ and $C, D$ respectively. If $\angle ACH = 35^\circ$, find $\angle HBD$.

In a circle with center $O$, arc $ABC$ subtends an angle of $140^\circ$ at the center. If $P$ is a point on the major arc $AC$, find the value of $\angle APC + \angle ABC$.

How many circles can pass through two distinct points?

In cyclic quadrilateral $ABCD$, if $\angle A = 5x$ and $\angle C = 4x$, find $\angle A$.

If all points of a polygon lie on a circle, the polygon is said to be...

Which of the following is a property of a cyclic quadrilateral?

In a cyclic quadrilateral $ABCD$, if $\angle B = 95^{\circ}$, then find the exterior angle at vertex $D$.

If $A, B, C, D$ are four points on a circle and $AC, BD$ intersect at $E$ such that $\angle BEC = 130^{\circ}$ and $\angle ECD = 20^{\circ}$, find $\angle BAC$.

In cyclic quadrilateral $PQRS$, if $\angle P = 5x + 5$ and $\angle R = 10x + 25$, find $x$.

Points $A, B, C, D$ lie on a circle. Chords $AB$ and $CD$ produced meet at $P$. If $\angle PBC = 70^\circ$ and $\angle PCB = 60^\circ$, find $\angle ADC$.

In a cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular. If $\angle BAC = 35^\circ$, find $\angle CAD$.

In $\triangle ABC$, $\angle A = 40^\circ$ and $\angle B = 60^\circ$. The circumcircle of $\triangle ABC$ is drawn. Find the angle subtended by the side $AB$ at the circumcenter.