Question 1: In the figure, given below, and are two parallel chords and is the center. If the radius of the circle is , find the distance between the two chords of lengths and respectively. **[2010]**

Answer:

We know that perpendicular drawn from the center of the circle will bisect the chord.

Therefore

Hence

Question 2: In the given figure and ; find the values of and . **[2007]**

Answer:

Given and

(ABDE is a cyclic quadrilateral)

(straight line)

In :

Therefore

Therefore is a cyclic quadrilateral)

In :

Question 3: In the following figure shows a circle with as its diameter. If and , find the perimeter of the cyclic quadrilateral .** [1992]**

Answer:

Given:

Therefore the perimeter of

Question 4: In the given circle with diameter , find the value of .** [2003]**

Answer:

(angles in the same segment)

In

Since is the diameter,

(angle in the semi circle)

Question 5: In the given diagram, , is a diameter of the circle. Calculate: (i) (ii) (iii) **[2014]**

Answer:

Given is diameter

(i) (angle in semi circle)

In

(ii) is a cyclic quadrilateral

(angle in the same segment)

Therefore

(iii) (angle in the same segment)

Question 6: In the given diagram, is the side of a regular hexagon, is the side of a regular pentagon and is a diameter. Calculate:

(i)

(ii)

(iii)

(iv) **[1984]**

Answer:

is a side of a regular hexagon

is a side of regular pentagon

(ii) In

(radius of the same circle)

(iii) In

(iv) In cyclic quadrilateral

(opposite angles of a cyclic quadrilateral are supplementary)

Question 7: In the diagram, is the center of the circle and the length of . If find:

(i)

(ii) **[1996]**

Answer:

Given

(i)

(ii) (angle subtended at the center is twice that subtended at the circumference by a chord)

(iii) Similarly,

In cyclic quadrilateral

Question 8: In the given diagram, and . Find:

(i)

(ii)

(iii) **[1993]**

Answer:

Given

(equal arcs subtend equal angles at the center of a circle)

(i) In cyclic quadrilateral

(opposite angles in a cyclic quadrilateral are supplementary)

Since (equal chords subtend equal angles on the circumference)

(ii) Since

(iii) subtends at the center and on the circumference

Question 9: In a regular pentagon inscribed in a circle, find the ratio of the **[1990]**

Answer:

Consider

It subtends at the center and at the circumference

Therefore

Similarly, for we have

Question 10: In the given figure, and .

(i) Prove that is a diameter of the circle.

(ii) find **[2013]**

Answer:

Given and .

(i) In

Therefore

Therefore is the diameter (Theorem 11)

(ii) (angles in the same segment)

Therefore since

Question 11: In each of the following figures, is the center of the circle. Find the values of . **[2007]**

(i) | (ii) |

(iii) | (iv) |

Answer:

(i) is the diameter

Since (angle in same segment) Therefore

(ii) (angle in same segment)

In ,

(iii) In

radius

(iv) Since is the diameter

(angles in the same segment)

Question 12: In the given figure and , calculate (i) (ii) **[1995]**

Answer:

(i)

(ii)

(angles in the same segment)

Question 13: In the given figure is the diameter of the circle with center . Chord . Write down the angles and in terms of . **[1996]**

Answer:

(angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

(angles in a semicircle)

Now (angles in the same segment)

Question 14: In the given figure is the diameter of the circle with center . . and . Calculate (i) (ii) (iii) **[1998]**

Answer:

(i) (Since is a straight line)

(angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

(ii)

(given)

(angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

(iii) (cyclic quadrilateral)

Question 15: In the given figure, is the diameter of the circle. Write down the numerical value of . Give reasons for your answer. **[1998]**

Answer:

is a cyclic quadrilateral

Similarly,

Question 16: In the given figure is the diameter and . If , calculate .** [1991]**

Answer:

(angle in a semi circle)

(angles in the same segment)

(alternate angles)

Question 17: Use the given figure below to find (i) (ii) **[1987]**

Answer:

(i) In

(ii)

Question 18: In the given figure is the diameter and . If , find in terms of , the values of: (i) (ii) (iii) (iv) **[1991]**

Answer:

(i) (angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

(ii) (alternate angles)

In

(radius of the same circle)

(iii) (angles subtended by an chord on the center is double that subtended by the same chord on the circumference)

(iv) (given)

(alternate angles)

Question 19: In the figure is the diameter of the circle with center . . Find (i) (ii) **[2012]**

Answer:

(i) is a cyclic quadrilateral)

(ii) In

Question 20: In the given figure is the diameter of the circle whose center is . Given , calculate . **[1992]**

Answer:

Join P and S as shown in the diagram.

(angle in a semi circle)

In

Question 21: In the given figure is the diameter. Chord . Given the , calculate (i) (ii) **[1989]**

Answer:

Join P and R as shown in the diagram.

(i) (angles in the semi circle)

(ii) (given)

(alternate angles)

( is a cyclic quadrilateral)

Question 22: is the diameter of the circle with center . and . Calculate the numerical values of (i) (ii) (iii) **[1987]**

Answer:

Join B and D as shown in the diagram.

(i) (angle subtended by an chord on the center is double that subtended by the same chord on the circumference)

(ii) (angle in semi circle)

Since is equilateral

Since

(alternate angles)

(ii)

( is a cyclic quadrilateral)

Question 23: In the given figure, the center of the smaller circle lies on the circumference of the bigger circle. If and , find: (i) (ii) (iii) (iv) . **[1984]**

Answer:

Join A and B as shown in the diagram.

(i) (angle subtended by an chord on the center is double that subtended by the same chord on the circumference)

(ii) ( is a cyclic quadrilateral)

(iii) ( is a cyclic quadrilateral)

(iv) ( is a cyclic quadrilateral)

Question 24: is the center of the circle of radius . is any point in the circle such that . is the point travelling along the circumference, is the distance from . What are the least and the greatest values of . What is the position of the points at these values . **[1992]**

Answer:

When is on , the least distance of

When is on extended , the greatest distance of

Question 25: In the given figure, is the center of the circle. and are two chords of circle. and . , . Find the

(i) radius of the circle

(ii) length of chord ** [2014]**

Answer:

, ,

Therefore

(i) Consider

Radius of the circle.

(ii) Now consider

Question 26: In the given figure is the center of the circle Tangent of and meet at if , find (i) (ii) (iii) **[2011]**

Answer:

Consider and

is common

(two tangents drawn from a point on a circle are of equal lengths)

Therefore

(i)

(ii)

(iii) (chord subtends twice the angle at the center than that it subtends on the circumference)

Question 27: In the following figure O is the center of the circle and AB is a tangent to it at point B. . Find .** [2010]**

Answer:

is a tangent to the circle.

(given)

(angle at the center)