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