Question 1

In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?

Question 2

An arc of a circle subtends an angle of 70° at the centre. What is the measure of the angle subtended by the arc at a point on the circle?

Question 3

The diameter of a circle is 26 cm. A chord of length 24 cm is drawn in the circle. Find the distance from the centre of the circle to the chord.

Question 4

A circle has a radius of 15 cm. A chord is drawn. The distance from the centre of the circle to the chord is 9 cm. What is the length of the chord?

Question 5

Prove that the perpendicular bisector of a chord passes through the centre of the circle.

Question 6

The diameter of a circle is AB. Point C is on the circumference. What is the measure of the ∠ACB? Explain your reasoning.

Question 7

ABCD is a cyclic quadrilateral inscribed in a circle. If ∠A measures 75°, what is the measure of ∠C? If ∠B measures 110°, what is the measure of ∠D?

Question 8

Quadrilateral PQRS is inscribed in a circle. If ∠P = (2x + 10)° and ∠R = (3x − 20)°, find the value of x and the measures of ∠P and ∠R.

Question 9

The distance of a chord of length 16 cm from the centre of a circle is 6 cm. Find the radius of the circle.

Question 10

A cyclic quadrilateral has sides 5, 5, 12, 12 units. Find its area.

Question 11 ★

Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out?

Question 12 ★

When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord.

Question 13 ★

Draw a circle in which a chord of 6 cm length stands at a distance of 3 cm from the centre.

(Hint: Is it a circumcircle of a suitable triangle?)

Question 14 ★

Show that rectangle is the only parallelogram that can be inscribed in a circle.

Question 15 ★

Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.

Question 16 ★

Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?

Question 17 ★

In a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: “The centre of the circle lies on the angle bisector of ∠BAC”.

Question 18

Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle.

Question 19 ★

A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.

Question 20

A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP? Explain your reasoning.

Question 21

Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., ∠CDE = ∠ABC, where E is a point on the extension of side CD).

Question 22 ★

“There is no chord of a circle that is longer than its diameter.” How do you justify this statement?

Question 23 ★

Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes through point A is the one that is perpendicular to OA.

Question 24

How would you use the following figure to justify the statement that the angle in a semicircle is 90°?

Question 25 ★

In a circle, two chords CC’ and DD’ are drawn perpendicular to a diameter AB. Prove that the segment MM’ joining the midpoints of the chords CD and C’ D’ is perpendicular to AB.

Question 26 ★

How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is 180°?