Practice Questions

Prove that if two minor arcs of a circle are congruent, their corresponding chords are equal.

Justify whether a rhombus that is not a square can be a cyclic quadrilateral.

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

State the theorem regarding the perpendicular from the centre to a chord, and also state its converse.

Define a cyclic quadrilateral.

State the theorem concerning angles that are in the same segment of a circle.

Formulate the specific condition that a parallelogram must satisfy to be cyclic.

Two equal chords AB and CD of a circle with center O subtend angles $\angle AOB = 75^\circ$ and $\angle COD = (5x - 5)^\circ$. Analyze the relationship and find the value of $x$.

In a circle with center O, an arc PQ subtends an angle $\angle PAQ = 48^\circ$ at a point A on the remaining part of the circle. Calculate the measure of the reflex angle $\angle POQ$.

In a circle with centre O, if PQ is a minor arc, what name is given to the angle subtended by the major arc PQ at the centre?

What is the measure of an angle in a semicircle?

A circle has a radius of 13 cm. A chord is drawn at a distance of 12 cm from the center. Calculate the length of this chord.

Prove that if a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals are also equal.

State the relationship between congruent arcs and their corresponding chords in a circle.

How is the distance of a chord from the centre of a circle defined?

What is the key property of the opposite angles of a cyclic quadrilateral? State the relevant theorem.

In a circle, points P, Q, R, and S are concyclic. Diagonals PR and QS intersect at T. If $\angle QPR = 40^\circ$ and $\angle PQS = 60^\circ$, analyze the angles in the same segment to find $\angle QRS$.

Two circles with centers O and O' intersect at points A and B. The radii are 10 cm and 17 cm respectively. If the length of the common chord AB is 16 cm, calculate the distance between their centers, OO'.

Describe the relationship between the length of a chord and the magnitude of the angle it subtends at the centre of the circle.

Explain the theorem that relates equal chords of a circle to their distances from the centre. Also, state its converse. Use simple descriptions of diagrams to illustrate both parts.

Evaluate whether any kite can be a cyclic quadrilateral. Justify your answer.

Explain the concept of an 'angle subtended by a line segment at a point'. Illustrate with a simple diagram.

Points A, B, C, and D are on a circle. If arc BCD subtends an angle of $130^\circ$ at the center, calculate the measure of $\angle BAD$.

In a cyclic quadrilateral PQRS, if $\angle P = (3x + 10)^\circ$ and $\angle R = (2x + 20)^\circ$, solve for $x$ and find the measure of $\angle P$ and $\angle R$.

In a circle with center O, chord AB is 24 cm long. If the radius of the circle is 15 cm, calculate the distance of the chord AB from the center O.

ABCD is a cyclic quadrilateral where AB is the diameter of the circle. If $\angle ADC = 125^\circ$, calculate the measure of $\angle BAC$.

A circle has a radius of 20 cm. Two parallel chords of lengths 32 cm and 24 cm are drawn. Calculate the distance between the chords if they lie on opposite sides of the center.

Critique the following statement: "If two chords of a circle are equidistant from the center, they must be parallel."

In a circle with center O, chords AB and CD are equal. If $\angle OAB = 55^\circ$, determine and justify the value of $\angle OCD$.

Two concentric circles with center O are given. A chord AB of the larger circle is tangent to the smaller circle at point P. Prove that P is the midpoint of the chord AB.

ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and $\angle ADC = 130^\circ$. Determine and justify the measure of $\angle BAC$.

Identify the three facts needed to prove $\triangle AOB \cong \triangle COD$ by the SSS rule, given that AB and CD are two equal chords of a circle with centre O.

Design a method to find the center of a given circle using only a straightedge and a compass. Justify each step of your construction.

Two equal chords AB and CD of a circle with center O intersect at a point E inside the circle. Prove that the line segment OE is equally inclined to the chords (i.e., it makes equal angles with the chords).

In the given figure, O is the center of the circle. If $\angle OPR = 25^\circ$ and $\angle OQR = 30^\circ$, calculate the measure of $\angle POQ$.

Justify why the perpendicular bisector of any chord of a circle must pass through its center.

Describe the condition required for four points to be concyclic. State the associated theorem and briefly explain the logic behind it.

Theorem 9.9 states: "If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle." Critique this theorem by constructing a scenario where a line segment AB subtends equal angles at points C and D, yet A, B, C, and D are not concyclic. Justify why your scenario does not contradict the theorem.

Two circles intersect at points P and Q. A straight line through P intersects the first circle at A and the second circle at B. Another straight line through Q intersects the first circle at C and the second circle at D. Prove that $AC$ is parallel to $BD$.

In a cyclic quadrilateral ABCD, the diagonal AC bisects $\angle BCD$. Given $\angle ABC = 108^\circ$ and $\angle CAD = 42^\circ$. Calculate the measure of $\angle BAC$ and $\angle ADC$.

PQ is a diameter of a circle with center O. If chord PR is drawn such that $\triangle POR$ is an equilateral triangle, calculate the length of chord QR if the radius is 6 cm.

Two chords AB and CD of a circle are parallel to each other. AB = 10 cm and CD = 24 cm. If the chords are on the same side of the center and the distance between them is 7 cm, calculate the radius of the circle.

A chord PQ of a circle is equal to its radius. Determine and justify the angle subtended by this chord at a point on the major arc and at a point on the minor arc.

Three friends Anjali, Brijesh, and Charan are sitting on the circumference of a circular park of radius 10 m. The distance between Anjali and Brijesh (AB) is 12 m, and the distance between Brijesh and Charan (BC) is also 12 m. Analyze the geometric arrangement to calculate the distance between Anjali and Charan (AC).

Summarize five fundamental properties related to chords and angles in a circle as stated in the chapter.