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)