Question 1: In the given diagram, and AB = AC.

(i) What is the relation between and ?

(ii) What is the relation between and ?

(iii) If is greater than , then what is the relation between the chord and ?

(iv) If , find the measure of .

Answer:

Given:

(i) (Since equal chords subtend equal arcs)

(ii) (equal arcs will subtend equal angles at the center)

(iii) If

Therefore

(iv) Given

Therefore

Question 2: In , the from vertices and on their opposite sides meet (when produced) the circumference of the triangle at points and respectively. Prove that:

Answer:

Consider &

(given)

(vertically opposite angles)

(AAA postulate)

Therefore

Therefore Arcs that subtend equal angle at circumference are equal

Therefore

Question 3: In a cyclic trapezium, prove that non parallel sides are equal and the diagonals are also equals.

Answer:

is a cyclic trapezium.

Chord subtends at circumference

Chord subtends at circumference

But (vertically opposite angles)

Therefore

Now in &

is common

(angles in the same segment)

(proved above)

(SAS axiom)

Question 4: In the given diagram, is the diameter of the circle with center . Chords . If , find (i) (ii)

Answer:

Given

(i) is the diameter

(angle in semicircle)

(ii) Since (given)

(equal arcs subtend equal angles at center)

We know

In

(radius of the same circle)

Therefore

is a cyclic quadrilateral

(opposite angles are supplementary)

Now

Question 5: In the given diagram, if of circle with center , prove that quadrilateral is an isosceles trapezium.

Answer:

Given

(Equal arcs subtend equal angles at the circumference)

(alternate angles are equal)

Therefore is a trapezium

Also

Therefore is an Isosceles trapezium.

Question 6: In a given figure is an isosceles triangle and is the center of the circumcircle. Prove bisects .

Answer:

Given (Since is an isosceles triangle)

(equal chords subtend equal angles on the circumference)

Therefore bisects .

Question 7: If two sides of a cyclic quadrilateral are parallel, prove that:

(i) its other two sides are equal

(ii) its diagonals are equal

Answer:

(i) Given

Therefore (alternate angles)

Chord subtends on circumference.

Chord subtends on circumference.

Therefore (since equal chords subtend equal angles on the circumference)

(ii) Consider

is common

(proved above)

(angles in the same segment)

Therefore (by SAS axiom)

Therefore (corresponding parts of congruent triangles are equal)

Question 8: In the given diagram, circle with center . and . Calculate:

(i)

(ii)

(iii)

Answer:

Given

(Equal chords subtend equal angles at the center of a circle)

(angle subtended at the center is twice that of the one subtended on the circumference)

In ,

(radius of the same circle)

Therefore

Similarly, in

and in

Therefore

(i)

(ii)

(iii)

Question 9: In the given diagram, is the side of regular six-side polygon and is a side of a regular eight-sides polygon inscribed in a circle with center . Calculate:

(i)

(ii)

(iii)

Answer:

(i) Since is the side of a regular hexagon

(ii) subtends at the circumference and at the center

We know

(iii) Since is the side of a regular octagon

subtends at the circumference and at the center

Therefore

Question 10: In a regular pentagon inscribed in a circle, find the ratio of the **[1990]**

Answer:

Consider

It subtends at the center and at the circumference

Therefore

Similarly, for we have

Question 11: In the given diagram, and . Find:

(i)

(ii)

(iii) **[1993]**

Answer:

Given

(equal arcs subtend equal angles at the center of a circle)

(i) In cyclic quadrilateral

(opposite angles in a cyclic quadrilateral are supplementary)

Since (equal chords subtend equal angles on the circumference)

(ii) Since

(iii) subtends at the center and on the circumference

Question 12: In the diagram, is the center of the circle and the length of . If find:

(i)

(ii) **[1996]**

Answer:

Given

(i)

(ii) (angle subtended at the center is twice that subtended at the circumference by a chord)

(iii) Similarly,

In cyclic quadrilateral

Question 13: In the diagram given below, is the center of the circle. is the side of regular pentagon and is the side of a regular hexagon. Find the angles of the .

Answer:

is the side of a regular pentagon

is a side of regular hexagon

Now

subtends at the center and on the circumference

Similarly

Therefore

Question 14: In the given diagram, is the side of a regular hexagon, is the side of a regular pentagon and is a diameter. Calculate:

(i)

(ii)

(iii)

(iv) **[1984]**

Answer:

is a side of a regular hexagon

(i)

is a side of regular pentagon

(ii) In

(radius of the same circle)

(iii) In

(iv) In cyclic quadrilateral

(opposite angles of a cyclic quadrilateral are supplementary)