Question 1: In the given figure,
is the center of the circle.
and
. Find
.
Answer:
In
Therefore
In
Therefore
We know (Theorem 9)
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)
Question 11: In the following figure,
is the center and
is equilateral. Find (i)
(ii)
Answer:
In , since it is equilateral, all angles are equal.
(angles int he same segment)
Since is a cyclic quadrilateral
Question 12: Given:
and
. Find the value of
.
Answer:
(angles in same segment)
Let (angles in same segment)
Let
Question 13:
is a cyclic quadrilateral in circle with center
. If
; find
.
Answer:
(angle in the semi circle)
Question 14: In the given figure,
is a diameter of the circle and
. Find
.
Answer:
(radius of the same circle)
Therefore in
(angle in the same segment)
Question 15: In the following figure,
is the center of the circle,
and
. Find
.
Answer:
Let
Hence
Question 16:
is a cyclic quadrilateral in which
and
. Calculate (i)
(ii)
(iii)
Answer:
(i)
(ii)
is a cyclic quadrilateral
Question 17: In the figure given below,
is the diameter of the circle whose center is
. Given that:
. Show that
.
Answer:
In
(radius of the same circle)
Hence
Question 18: In the figure given below,
and
are straight lines through the center
of the circle. If
and
find (i)
(ii)
Answer:
(angle in a semi circle)
(i) Therefore
Question 19: In the given figure,
is a diameter of a circle whose center is
. A circle is described on
as a diameter.
is a chord of the larger circle intersects the smaller circle at
. Prove
.
Answer:
(angles in a semi circle)
(common angle)
Hence
Question 20: In the following figure, (i) if
, find
and
. (ii) Prove that
is parallel to
.
Answer:
(i) is a cyclic quardilateral
(ii) Since
Question 21: Prove that: (i) the parallelogram inscribed in a circle is a rectangle (ii) the rhombus, inscribed in a circle is a square.
Answer:
(i) Let be a parallelogram inscribed in the circle.
(opposite angles of a parallelogram are equal)
Similarly,
Therefore is a rectangle
(ii) is a rhombus (given) i.e. all four sides are equal.
Similarly,
Therefore is a square
Question 22: In the following figure
. Prove that
is an isosceles trapezium.
Answer:
(given)
is a cyclic quadrialteral)
… … … … (i)
(corresponding angles)
Hence
… … … … (ii)
Hence because of (i) and (ii), is an isosceles trapezium.
Question 23: Two circles intersect at and
. Through
diameters
and
of the two circles are drawn. Show that the points
and
are collinear.
Answer:
(angle in a semi circle)
(angle in a semi circle)
Therefore
Therefore are collinear.
Question 24: is a quadrilateral inscribed in a circle. having
.
is the center of the circle. Show that:
.
Answer:
(cyclic quadrilateral)
(sum of the angles in a triangle is 180)
Similarly
Therefore
Question 25: The figure given below shows the circle with center
. Given
and
. (i) Find the relationship between
. (ii) Find
if
is a parallelogram.
Answer:
(i) and
(given)
(ii) If is a parallelogram
Therefore (opposite angles are equal)
(radius of the same circle)
Hence
Question 26: Two chords and
intersect at
inside the circle. Prove that the sum of the angles subtended by the arcs
and
at the center
is equal to twice the
.
Answer:
Similarly,
Adding the two
… … … … (i)
In
… … … … (ii)
Using (1) and (ii)
Question 27: In the given figure
is a diameter of the circle.
and
. Find (i)
(ii)
Answer:
(i) (angle in a semi circle)
(ii) Given
(alternate angles)
Question 28: In the given figure,
and
is the center of the circle. If
, find the
. Give reasons.
Answer:
and
(angles in a semi circle)
Since
In
(cyclic quadrilateral)
(angles in the same segment)
Question 29: Two circles intersect at
and
. Through
a straight line
is drawn to meet the circles in
and
. Through
, a straight line is drawn to meet the circles at
and
. Prove that
is parallel to
.
Answer:
and
are cyclic quadrilateral
… … … … (i)
… … … … (ii)
… … … … (iii)
From (i) and (ii)
… … … … (iv)
From (iv) and (iii)
… … … … (v)
Therefore by (v)
Question 30: is a cyclic quadrilateral in which
and
on being produced, meet at
such that
. Prove that
is parallel to
.
Answer:
is a cyclic quadrilateral,
and
(given)
Question 31:
is a diameter of a circle,
as shown in the figure.
and
are straight lines. Find (i)
(ii)
(iii)
Answer:
(i) (angles in the same segment of the circle subtended by the same chord)
(ii)
(angle in a semi circle subtended by the diameter)
(iii)
Question 32: In the given figure,
is bisector of
and
is a cyclic quadrilateral. Prove that:
.
Answer:
is a cyclic quadrilateral
(cyclic quadrilateral)
Also (angles in the same segment of the circle subtended by the same chord)
Question 33: In the figure,
is the center of the circle,
. Calculate
and
.
Answer:
Question 34: In the given figure,
and
are the centers of two intersecting circles intersecting at
and
.
is a straight line. Calculate the numerical value of
.
Answer:
is a straight line
Question 35: In the figure given below, two circles intersect at
and
. The center of the smaller circle is
and lines on the circumference of the larger circle. Given
. Find in terms of
the value of (i) Obtuse
(ii)
(iii)
. Give reasons.
Answer:
(i)
(ii) is a cyclic quadrilateral)
(iii) (angles in the same segment of the circle subtended by the same chord)
Question 36: In the given figure
is the cent of the circle and
. Calculate
and
.
Answer:
is a cyclic quadrilateral
Question 37: In the given figure,
is the center of the circle,
is a parallelogram and
is a straight line. Prove that
.
Answer:
(angle subtended at the center is twice subtended on the circumference by the same chord)
(alternate angles)
is a parallelogram
(opposite angles in a parallelogram are equal)
Question 38:
is a cyclic quadrilateral in which
is parallel to
and
is a diameter of the circle. Given
; calculate: (i)
(ii)
.
Answer:
(angles in the same segment of the circle subtended by the same chord)
(angle subtended by the diameter on a semi circle)
(alternate angles)
Question 39: In the given figure
is the diameter of the circle. Chord
and
. Calculate (i)
(ii)
.
Answer:
(i) (angle in the semi circle)
(ii)
(alternate angles)
is a cyclic quadrilateral
Question 40: The sides
and
of a cyclic quadrilateral
are produced to meet at
, the sides
and
are produced to meet at
. If
and
find (i)
(ii)
Answer:
(vertically opposite angles)
(cyclic quadrilateral)
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)
(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:
(angles subtended by an chord on the center is double that subtended by the same chord on the circumference)
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)
Question 51: In the given figure
is the diameter and
. If
, find in terms of
, the values of: (i)
(ii)
(iii)
(iv)
[1991]
Answer:
(i) (angles subtended by an chord on the center is double that subtended by the same chord on the circumference)
(ii) (alternate angles)
In
(radius of the same circle)
(iii) (angles subtended by an chord on the center is double that subtended by the same chord on the circumference)
(iv) (given)
(alternate angles)
Question 52: In the figure
is the diameter of the circle with center
.
. Find (i)
(ii)
[2012]
Answer:
(i) is a cyclic quadrilateral)
(ii) In
Question 53: In the given figure
is the diameter of the circle whose center is
. Given
, calculate
. [1992]
Answer:
Join P and S as shown in the diagram.
(angle in a semi circle)
In
Question 54: In the given figure
is the diameter. Chord
. Given the
, calculate (i)
(ii)
[1989]
Answer:
Join P and R as shown in the diagram.
(i) (angles in the semi circle)
(ii) (given)
(alternate angles)
(
is a cyclic quadrilateral)
Question 55:
is the diameter of the circle with center
.
and
. Calculate the numerical values of (i)
(ii)
(iii)
[1987]
Answer:
Join B and D as shown in the diagram.
(i) (angle subtended by an chord on the center is double that subtended by the same chord on the circumference)
(ii) (angle in semi circle)
Since is equilateral
Since
(alternate angles)
(ii)
(
is a cyclic quadrilateral)
Question 56: In the given figure, the center
of the smaller circle lies on the circumference of the bigger circle. If
and
, find: (i)
(ii)
(iii)
(iv)
. [1984]
Answer:
Join A and B as shown in the diagram.
(i) (angle subtended by an chord on the center is double that subtended by the same chord on the circumference)
(ii) (
is a cyclic quadrilateral)
(iii) (
is a cyclic quadrilateral)
(iv) (
is a cyclic quadrilateral)
Question 57: In the figure,
,
and
. Find (i)
(ii)
. Hence show that
is the diameter.
Answer:
(i) (
is a cyclic quadrilateral)
(ii) In
(angles in the same segment)
Therefore is the diameter since the angle subtended by
on the circumference is
.
Question 58: In a cyclic quadrilateral ,
and
. Find each angle of the quadrilateral.
Answer:
(
is a cyclic quadrilateral)
Also given
(
is a cyclic quadrilateral)
Question 59: The given figure shows a circle with center
and
. Find (i)
(ii)
Answer:
Join A and P as shown in the diagram.
(i) (angle in semi circle)
Now, (angles in the same segment)
(ii) in
Question 60: In the given figure
is the center of the circle. Chords
and
are perpendicular to each other. If
and
(i) express
in terms of
(ii) express
in terms of
(iii) prove that
Answer:
Given:
(i) In
In
(ii) (angle subtended by an chord on the center is double that subtended by the same chord on the circumference)
in
Now, (angles in the same segment)
(iii) and we proved that
Question 61: In a circle, with center , a cyclic quadrilateral
is drawn with
as a diameter of the circle and
equal to radius of the circle. If
and
produced meet at point
, show that
.
Answer:
In
is equilateral
(ABCD is a cyclic quadrilateral)
In