Question 1: In the given circle with diameter , find the value of .

Answer:

(angles in the same segment)

In

Since is the diameter,

(angle in the semi circle)

Question 2: In the given figure, is the center of the circle with radius . and are to and respectively. and . Determine the length of .

Answer:

Given:

are mid point of respectively ( from the center to a chord will bisect the chord)

In

Similarly, in

Hence

Question 3: The given figure shows two circles with centers and ; and radius and respectively, touching each other internally. If the perpendicular bisector of meets the bigger circle in and , find the length of .

Answer:

bisects (let the point be )

In

Question 4: In the given figure, in which . Show that is equal to the radius of the circumcircle of the , whose center is .

Answer:

Given

(angle at the center of the circle is twice that of the angle subtend at the circumference by the same chord)

In

  (radius of the same circle)

Therefore

Therefore is equilateral

Hence is equal to the radius of the circle.

Question 5: Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.

Answer:

Given: Isosceles , where .

is the diameter of the circle.

(angle in the semi circle)

Consider and

is common

Therefore 

Therefore

Hence is the midpoint of .

Question 6: In the given figure, chord is parallel to diameter of the circle. Given , calculate .

Answer:

Given

subtends at center  and at circumference.

In ,

(radius of the same circle)

Given

Question 7: Chords and of a circle intersect each other at point such that . Show that: .

Answer:

Given and intersect at

(given)

Therefore

Therefore

Question 8: The quadrilateral formed by the angle bisectors of a Cyclic Quadrilateral is also cyclic. Prove it.

Answer:

Given: is a cyclic quadrilateral and is a quadrilateral

In :

… … … … … (i)

In

  … … … … … (ii)

Adding (i) and (ii)

Therefore is a cyclic quadrilateral as the opposite angles are supplementary.

Question 9: In the given diagram, , is a diameter of the circle. Calculate: (i) (ii) (iii)

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 10: and are points on equal sides and of an isosceles such that . Prove that the points and are concyclic.

Answer:

In . Also given

Therefore

In

Now in

Therefore

is a cyclic quadrilateral.

Question 11: In the given figure, is a cyclic quadrilateral. and is produced to point . If ; Determine . Give reason in support of your answer.

Answer:

Given: is a cyclic quadrilateral

and

Since

(alternate angles)

Therefore

Therefore

Question 12: If is the incentre and when produced meets the circumcircle of at point . If and . Calculate (i) , (ii) , (iii)

Answer:

Given and

(i) (angles in the same segment)

Since is the incenter

Therefore

(ii) Similarly,

(iii) In

Also

Question 13: ln the given figure, and . Determine, in terms of : (i) , (ii) . Hence or otherwise, prove that .

Answer:

Given