Question 1: In the given figure, is the center of the circle. and . Find .

Answer:

In

Therefore

In

Therefore

We know (Theorem 9)

Hence

Question 2: 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 3: Given is the center of the circle and . Calculate the value of (i) (ii)

Answer:

radius

Therefore

In

radius

Therefore

In

(i)

(ii)

Question 4: In each of the following figures, is the center of the circle. Find the values of .

(i) | (ii) |

Answer:

(i)

Hence

(ii)

Question 5: 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 6: In the figure, is common chord of the two circles. If are diameters, prove that are in straight line. are the centers of the two circles.

Answer:

Since is the diameter

Also since is the diameter

Therefore is a straight line and hence are collinear.

Question 7: In the figure given below, find (i) (ii) (iii)

Answer:

is a cyclic quadrilateral

Also

Question 8: In the given figure, is the center of the circle. If and . Find (i) (ii) (iii) (iv)

Answer:

Since (radius of the same circle)

In

Question 9: Calculate (i) (ii) (iii)

Answer:

(BC subtends same angle in the same segment)

(AC subtends same angle in the same segment)

Question 10: In the figure given below, is a cyclic quadrilateral in which , and . Find: (i) (ii) (iii)

Answer:

(i)

(ii) (cyclic quadrilateral)

(iii)

Question 11: In the following figure, is the center and is equilateral. Find (i) (ii)

Answer:

In , since it is equilateral, all angles are equal.

(angles int he same segment)

Since is a cyclic quadrilateral

Question 12: Given: and . Find the value of .

Answer:

(angles in same segment)

Let (angles in same segment)

Let

Question 13: is a cyclic quadrilateral in circle with center . If ; find .

Answer:

(angle in the semi circle)

Question 14: In the given figure, is a diameter of the circle and . Find .

Answer:

(radius of the same circle)

Therefore in

(angle in the same segment)

Question 15: In the following figure, is the center of the circle, and . Find .

Answer:

Let

Hence

Question 16: is a cyclic quadrilateral in which and . Calculate (i) (ii) (iii)

Answer:

(i)

(ii)

is a cyclic quadrilateral

Question 17: In the figure given below, is the diameter of the circle whose center is . Given that: . Show that .

Answer:

In

(radius of the same circle)

Hence

Question 18: In the figure given below, and are straight lines through the center of the circle. If and find (i) (ii)

Answer:

(angle in a semi circle)

(i) Therefore

Question 19: In the given figure, is a diameter of a circle whose center is . A circle is described on as a diameter. is a chord of the larger circle intersects the smaller circle at . Prove .

Answer:

(angles in a semi circle)

(common angle)

Hence

Question 20: In the following figure, (i) if , find and . (ii) Prove that is parallel to .

Answer:

(i) is a cyclic quardilateral

(ii) Since

Question 21: Prove that: (i) the parallelogram inscribed in a circle is a rectangle (ii) the rhombus, inscribed in a circle is a square.

Answer:

(i) Let be a parallelogram inscribed in the circle.

(opposite angles of a parallelogram are equal)

Similarly,

Therefore is a rectangle

(ii) is a rhombus (given) i.e. all four sides are equal.

Similarly,

Therefore is a square

Question 22: In the following figure . Prove that is an isosceles trapezium.

Answer:

(given)

is a cyclic quadrialteral)

… … … … (i)

(corresponding angles)

Hence

… … … … (ii)

Hence because of (i) and (ii), is an isosceles trapezium.

Question 23: Two circles intersect at and . Through diameters and of the two circles are drawn. Show that the points and are collinear.

Answer:

(angle in a semi circle)

(angle in a semi circle)

Therefore

Therefore are collinear.

Question 24: is a quadrilateral inscribed in a circle. having . is the center of the circle. Show that: .

Answer:

(cyclic quadrilateral)

(sum of the angles in a triangle is 180)

Similarly

Therefore

Question 25: The figure given below shows the circle with center . Given and . (i) Find the relationship between . (ii) Find if is a parallelogram.

Answer:

(i) and (given)

(ii) If is a parallelogram

Therefore (opposite angles are equal)

(radius of the same circle)

Hence

Question 26: Two chords and intersect at inside the circle. Prove that the sum of the angles subtended by the arcs and at the center is equal to twice the .

Answer:

Similarly,

Adding the two

… … … … (i)

In

… … … … (ii)

Using (1) and (ii)

Question 27: In the given figure is a diameter of the circle. and . Find (i) (ii)

Answer:

(i) (angle in a semi circle)

(ii) Given

(alternate angles)

Question 28: In the given figure, and is the center of the circle. If , find the . Give reasons.

Answer:

and (angles in a semi circle)

Since

In

(cyclic quadrilateral)

(angles in the same segment)

Question 29: Two circles intersect at and . Through a straight line is drawn to meet the circles in and . Through , a straight line is drawn to meet the circles at and . Prove that is parallel to .

Answer:

and are cyclic quadrilateral

… … … … (i)

… … … … (ii)

… … … … (iii)

From (i) and (ii)

… … … … (iv)

From (iv) and (iii)

… … … … (v)

Therefore by (v)

Question 30: is a cyclic quadrilateral in which and on being produced, meet at such that . Prove that is parallel to .

Answer:

is a cyclic quadrilateral, and (given)

Question 31: is a diameter of a circle, as shown in the figure. and are straight lines. Find (i) (ii) (iii)

Answer:

(i) (angles in the same segment of the circle subtended by the same chord)

(ii)

(angle in a semi circle subtended by the diameter)

(iii)

Question 32: In the given figure, is bisector of and is a cyclic quadrilateral. Prove that: .

Answer:

is a cyclic quadrilateral

(cyclic quadrilateral)

Also (angles in the same segment of the circle subtended by the same chord)

Question 33: In the figure, is the center of the circle, . Calculate and .

Answer:

Question 34: In the given figure, and are the centers of two intersecting circles intersecting at and . is a straight line. Calculate the numerical value of .

Answer:

is a straight line

Question 35: In the figure given below, two circles intersect at and . The center of the smaller circle is and lines on the circumference of the larger circle. Given . Find in terms of the value of (i) Obtuse (ii) (iii) . Give reasons.

Answer:

(i)

(ii) is a cyclic quadrilateral)

(iii) (angles in the same segment of the circle subtended by the same chord)

Question 36: In the given figure is the cent of the circle and . Calculate and .

Answer:

is a cyclic quadrilateral

Question 37: In the given figure, is the center of the circle, is a parallelogram and is a straight line. Prove that .

Answer:

(angle subtended at the center is twice subtended on the circumference by the same chord)

(alternate angles)

is a parallelogram

(opposite angles in a parallelogram are equal)

Question 38: is a cyclic quadrilateral in which is parallel to and is a diameter of the circle. Given ; calculate: (i) (ii) .

Answer:

(angles in the same segment of the circle subtended by the same chord)

(angle subtended by the diameter on a semi circle)

(alternate angles)

Question 39: In the given figure is the diameter of the circle. Chord and . Calculate (i) (ii) .

Answer:

(i) (angle in the semi circle)

(ii)

(alternate angles)

is a cyclic quadrilateral

Question 40: The sides and of a cyclic quadrilateral are produced to meet at , the sides and are produced to meet at . If and find (i) (ii)

Answer:

(vertically opposite angles)

(cyclic quadrilateral)

Question 41: In the given figure, is the diameter of the circle with center . and . Calculate (i) (ii) (iii) (iv) . Show that is an equilateral triangle.

Answer:

(i) is a cyclic quadrilateral

(ii) (angle in the semicircle)

In ,

In

(iii)

(alternate angles)

(iv)

(cyclic quadrilateral)

In ,

(Radius of the same circle)

Therefore all angles are . Hence is an equilateral triangle.

Question 42: In the given figure is the incenter of the . when produced meets the circumcenter of the the at . Given and . Calculate: (i) (ii) (iii) (iv)

Answer:

(i) is bisector of

(ii) (angles in the same segment)

(iii)

(iv)

bisects

Question 43: A is inscribed in a circle. The bisector of and meet the circumference of the triangle at points and respectively. Prove that (i) (ii) (iii)

Answer:

(i) bisects

(angle in same segment)

… … … … (i)

(ii) bisects

and (angles in the same segment)

… … … … (ii)

(iii) Adding (i) and (ii) we get

Question 44: Calculate angles if:

Answer:

Similarly

Therefore

Question 45: In the given figure and , calculate (i) (ii) **[1995]**

Answer:

(i)

(ii)

(angles in the same segment)

Question 46: 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 47: In the given figure is the diameter of the circle with center . $CD \parallel BE &s=0$. 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 48: 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 49: 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 50: Use the given figure below to find (i) (ii) **[1987]**

Answer:

(i) In

(ii)

Question 51: 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 52: In the figure is the diameter of the circle with center . . Find (i) (ii) **[2012]**

Answer:

(i) is a cyclic quadrilateral)

(ii) In

Question 53: 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 54: 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 55: 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 56: 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 57: In the figure, , and . Find (i) (ii) . Hence show that is the diameter.

Answer:

(i) ( is a cyclic quadrilateral)

(ii) In

(angles in the same segment)

Therefore is the diameter since the angle subtended by on the circumference is .

Question 58: In a cyclic quadrilateral , and . Find each angle of the quadrilateral.

Answer:

( is a cyclic quadrilateral)

Also given

( is a cyclic quadrilateral)

Question 59: The given figure shows a circle with center and . Find (i) (ii)

Answer:

Join A and P as shown in the diagram.

(i) (angle in semi circle)

Now, (angles in the same segment)

(ii) in

Question 60: In the given figure is the center of the circle. Chords and are perpendicular to each other. If and (i) express in terms of (ii) express in terms of (iii) prove that

Answer:

Given:

(i) In

In

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

in

Now, (angles in the same segment)

(iii) and we proved that

Question 61: In a circle, with center , a cyclic quadrilateral is drawn with as a diameter of the circle and equal to radius of the circle. If and produced meet at point , show that .

Answer:

In

is equilateral

(ABCD is a cyclic quadrilateral)

In