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)