Question 1: In the given circle with diameter , find the value of .** [2003]**

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) **[2014]**

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

(angles in the same segment)

Since

Similarly, ( )

In

Since

Therefore and hence (alternate angles)

Question 14: In the given figure; and are straight lines. Show that and are supplementary.

Answer:

Given are straight lines.

In cyclic quadrilateral

… … … … … (i)

Similarly in cyclic quadrilateral

… … … … … (ii)

Adding (i) and (ii)

Hence they are supplementary.

Question 15: In the given figure, is the diameter of the circle with center . If , find .

Answer:

subtends at the center and at the circumference of the circle.

Question 16: In a cyclic-quadrilateral . Sides and produced meet at point whereas sides and produced meet at point . lf ; find and .

Answer:

Given is a cyclic quadrilateral.

or if , then

(sum of the opposite angles in a cyclic quadrilateral is )

In

… … … (i)

In

Therefore … … … (ii)

Hence

Therefore

Question 17: In the following figure, is the diameter of a circle with center and is the chord with length equal to radius . If produced and produced meet at point ; show that .

Answer:

Given: is diameter,

In

is equilateral

Therefore

In

(radius of the same circle)

Therefore

Similarly in

(radius of the same circle)

Therefore

Since is cyclic quadrilateral

Therefore (opposite angles of a cyclic quadrilateral are supplementary)

In

. Hence proved.

Question 18: In the following figure, is a cyclic quadrilateral in which is parallel to . If the bisector of meets at point and the given circle at point , prove that: (i) (ii)

Answer:

Given: is a cyclic quadrilateral

is the angle bisector) … … … (i)

(i) (alternate angles) … … … (ii)

In

Using (i) and (ii) we get

(vertically opposite angles)

is a cyclic quadrilateral)

Also

(ii) subtends on circumference

subtends on circumference

Given

Therefore (equal arcs subtends equal angles on circumference)

Question 19: is a cyclic quadrilateral. Sides and produced meet at point ; whereas sides and produced meet at point . If , find the angles of the cyclic quadrilateral .

Answer:

Given

In

In

or

Therefore

Question 20: In the following figure shows a circle with as its diameter. If and , find the perimeter of the cyclic quadrilateral .** [1992]**

Answer:

Given:

Therefore the perimeter of

Question 21: In the following figure, is the diameter of a circle with center . If , prove that: (i) (ii) is bisector of . Further, if the length of , find : (a) (b) .

Answer:

Given:

Consider and

is common

(angles in semi circle)

Therefore

(i) (corresponding parts of congruent triangles)

(ii) (equal chords subtend equal angles on the circumference of the same circle)

Therefore bisects

Question 22: In cyclic quadrilateral ; and ; find: (i) (ii) (iii) (iv)

Answer:

Given and

(angles in the same segment)

(angles in the same segment)

(opposite angles in a cyclic quadrilateral are supplementary)

Therefore

Question 23: In the given figure and ; find the values of and . **[2007]**

Answer:

Given and

(ABDE is a cyclic quadrilateral)

(straight line)

In :

Therefore

Therefore is a cyclic quadrilateral)

In :

Question 24: In the given figure, and . Find (i) (ii) (iii)

Answer:

Given and . Also

Therefore (alternate angles)

Hence

Therefore

and

Therefore

In :

Therefore

(angles in the same segment)

Question 25: is a cyclic quadrilateral of a circle center such that is a diameter of the circle and the length of the chord is equal to the radius of the circle. If and produced meet at , show that .

Answer:

Given

In :

In (Radius of the same circle)

Let

Therefore … … … (i)

In (Radius of the same circle)

Let

Therefore … … … (ii)

Now, (straight line angle)

In :

Question 26: In the figure, given alongside, bisects . Show that bisects

Answer:

Given bisects

Therefore

(angles in the same segment)

(angles in the same segment)

Therefore

Hence

Question 27: In the figure shown, and . Find (i) (ii) (iii) (iv)

Answer:

From ( angles in the same segment)

From (angles in the same segment)

In

Since

In

is a cyclic quadrilateral.

Hence

(i)

(ii)

(iii)

(iv)

Question 28: In the figure, given below, and are two parallel chords and is the center. If the radius of the circle is , find the distance between the two chords of lengths and respectively. **[2010]**

Answer:

We know that perpendicular drawn from the center of the circle will bisect the chord.

Therefore

Hence