Practice Questions

Define a circle using the concept of a locus.

List the two main types of symmetry a circle possesses.

What is a cyclic quadrilateral?

An arc of a circle subtends an angle of $88^\circ$ at the centre. Calculate the angle subtended by it at any point on the remaining part of the circle.

Formulate a rule to determine if a parallelogram is cyclic, and justify your reasoning.

A circle has a radius of $14.5$ cm. Calculate the length of its longest chord.

A student claims that if two chords in two different congruent circles are equal, they must subtend equal angles at their respective centres. Justify this claim.

In a cyclic quadrilateral $PQRS$, if $\angle Q = 95^\circ$, calculate the measure of $\angle S$.

Justify why a regular pentagon inscribed in a circle must have all its sides equidistant from the centre of the circle.

Design a method using only a compass and a straightedge to find the centre of a given circle. Justify the steps of your construction.

What is the name of the longest chord in a circle?

Describe the location of the circumcentre for an acute-angled, an obtuse-angled, and a right-angled triangle.

A, B, and C are three points on a circle with centre O. The angles subtended by the chords AB and AC at the centre O are $\angle AOB = 90^\circ$ and $\angle AOC = 110^\circ$ respectively. Formulate a method to find $\angle BAC$ and evaluate the two possible values.

Explain why the angle in a semicircle is always $90^\circ$. Use the theorem about the angle subtended by an arc at the centre.

Describe the locus of the centres of all circles that can be drawn through two given points A and B. Explain this concept and identify the circle with the smallest possible radius.

State the relationship between the angle subtended by an arc at the centre and at a point on the remaining part of the circle.

Summarize the properties of a cyclic quadrilateral. State the main theorem regarding its opposite angles and the converse of this theorem.

Explain why there is only one unique circle that can pass through three non-collinear points.

State the theorem concerning equal chords of a circle and the angles they subtend at the centre. Also, state its converse.

Explain the relationship between a line drawn from the centre of a circle to the midpoint of a chord. Also, state the converse of this theorem.

Summarize the property that connects the lengths of two unequal chords to their distances from the centre of the circle.

$AB$ is a diameter of a circle and $C$ is a point on the circumference. If $\angle CAB = 35^\circ$, calculate the measure of $\angle ABC$.

A chord of length $30$ cm is drawn in a circle at a distance of $8$ cm from the centre. Calculate the radius of the circle.

The radius of a circle is $20$ cm. Calculate the length of a chord that is at a distance of $12$ cm from the centre.

In a circle with centre $O$, points $A$, $B$, and $C$ are on the circle such that $\angle BOC = 110^\circ$ and $\angle AOB = 90^\circ$. Determine the measure of $\angle BAC$.

$PQRS$ is a cyclic quadrilateral where $\angle P = (x+15)^\circ$ and $\angle R = (4x+15)^\circ$. Solve for $x$ and determine the measures of $\angle P$ and $\angle R$.

Two parallel chords are drawn in a circle of radius $13$ cm. If the chords have lengths $10$ cm and $24$ cm, and lie on opposite sides of the centre, calculate the distance between them.

Evaluate if it is possible to draw a circle passing through the four vertices of a kite that is not a rhombus. Justify your answer.

Critique the statement: "Any quadrilateral where the perpendicular bisectors of all four sides are concurrent must be a cyclic quadrilateral."

Prove that of all chords that can be drawn through a given point P inside a circle, the one that is shortest is the chord that is perpendicular to the diameter passing through P.

Prove that if two circles intersect at two distinct points, the line joining their centres is the perpendicular bisector of their common chord.

Justify that a trapezium is cyclic if and only if it is an isosceles trapezium.

A chord of a circle is equal to its radius. Formulate and prove a theorem about the angle subtended by this chord at any point on the major arc.

Analyze the given statement: A quadrilateral $ABCD$ has angles $\angle A = 70^\circ$, $\angle B = 100^\circ$, $\angle C = 110^\circ$, and $\angle D = 80^\circ$. Determine if this quadrilateral can be a cyclic quadrilateral.

In a circle with centre O, chord $AB$ is equal to chord $AC$. If $\angle BAC = 80^\circ$, analyze the triangles formed and calculate $\angle OBC$.

Two parallel chords of lengths $12$ cm and $16$ cm are drawn in a circle of radius $10$ cm. If the chords are on the same side of the centre, analyze the geometry and calculate the distance between the two chords.

Points A, B, C, and D are on a circle. AC and BD are chords intersecting at point E. If $\angle DAC = 30^\circ$ and $\angle AEB = 85^\circ$, analyze the angles and calculate $\angle ABC$.

Design and justify a method to construct a triangle ABC, given the length of its base BC, the vertical angle $\angle A = \alpha$, and the length of the altitude from vertex A to the base BC, say $h$.

Explain the relationship between the length of a chord and its distance from the centre. Your explanation should cover both equal and unequal chords.

Two equal chords AB and CD of a circle with centre O intersect at a point P inside the circle. Prove that the line segment OP bisects the angle formed by the chords, $\angle APD$.

In a circle with centre O, if $\angle OAB = 40^\circ$ and $\angle OCB = 30^\circ$, calculate the measure of the reflex angle $\angle AOC$. (A, B, C are points on the circle).

Two circles with centres O1 and O2 touch each other externally at point P. A direct common tangent is drawn touching the circles at points A and B respectively. Propose and prove a theorem about the measure of angle $\angle APB$.

A regular hexagon with side length $6$ cm is inscribed in a circle. Analyze the properties of the hexagon and calculate the radius of the circle and the distance of each side from the centre.

State the theorem that provides the condition for four points to be concyclic, based on the angles subtended by a line segment.

In a circle with centre O, chord AB is produced to a point P such that the length of the segment BP is equal to the radius of the circle. The line from P passes through the centre O and intersects the circle at C and D. Prove that $\angle AOD = 3 \times \angle APC$.