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