Question 1: A chord of length is drawn in a circle of radius
. Calculate its distance from the center of the circle.
Answer:
Let the distance from the center
Therefore
Question 2: A chord of length is drawn at a distance of
from the center of a circle. Calculate the radius of the circle.
Answer:
Let the radius
Therefore
Question 3: The radius of a circle is and the length of perpendicular drawn from the center to a chord is
. Calculate the length of the chord.
Answer:
Let the length of the chord
Therefore
Therefore the length of the chord
Question 4: A chord of length 24 cm is at a distance of 5 cm from the center of the circle. Find the length of the chord of the same circle which is at a distance of 12 cm from the center.
Answer:
Let the radius
Therefore
Let the length of the chord
Therefore
Therefore the length of the chord
Question 5: In the following figure,
is a straight line
and
is the center of both the circles. If
and
, find the length of
.
Answer:
Therefore
Question 6: is the center of the circle of radius
.
is any point in the circle such that
.
is the point travelling along the circumference,
is the distance from
. What are the least and the greatest values of
. What is the position of the points
at these values . [1992]
Answer:
When is on
, the least distance of
When is on extended
, the greatest distance of
Question 7: In a circle of radius , two parallel chords of length
and
are drawn. Find the distance between the chords, if both the chord are:
(i) on the opposite side of the center
(ii) on the same side of the center
Answer:
(i) When the chords are on the opposite side of the triangles
Distance of the larger chord from the center
Distance of the smaller chord from the center
Therefore the distance between the chords
(ii) When the chords are on the same side of the triangles
Distance of the larger chord from the center
Distance of the smaller chord from the center
Therefore the distance between the chords
Question 8: Two parallel chords are drawn in a circle of diameter . The length of one chord is
. and the distance between the two chords is
. Find the length of another chord.
Answer:
Let the distance of the chord from the center
Therefore
Therefore the distance of the other chord from the center
Therefore the length of the other chord
Question 9: A chord
of a circle whose center is
, is bisected at
by a diameter
. Given
and
, calculate the lengths of (i)
(ii)
and (iii)
Answer:
Question 10: The figure given below, shows a circle with center
in which Diameter
bisects the chord
at point
. If
and
, find the radius of the circle.
Answer:
Let
Therefore
Therefore Radius
Question 11: The figure shows two concentric circles and
is the chord of larger circle. Prove that
.
Answer:
If you drop a perpendicular from to the chord, it would bisect the chord (Theorem 5: A perpendicular to a chord, from the center of the circle, bisects the chord.). Let us say that it intersects
and
at
.
Therefore and
If you subtract these two, we get
Question 12: A straight line is drawn cutting two equal circles and passing through the midpoint
of the line joining their centers
. Prove that the chords
, which are intercepted by two circles are equal.
Answer:
First draw perpendiculars and
Now consider and
(opposite angles)
is the mid point of
– Given)
Therefore
Therefore
Therefore (Theorem 8 (Converse of Theorem 7) : Chords of circle which are equidistant from the center of the circle are equal in length.)
Question 13:
are mid points of two equal chords
respectively of a circle with center
. Prove that:
(i)
(ii)
Answer:
First drop perpendiculars and
on
and
respectively.
will bisect
and
will bisect
. (Theorem 5: A perpendicular to a chord, from the center of the circle, bisects the chord.)
Therefore
Given . Therefore
Also
Therefore in (since angles opposite equal sides of a triangle are equal)
We know
(i) Therefore
(ii) We know
Therefore
Question 14: In the following figure;
and
are the points of intersection of two circles with centers
and
. If straight lines
and
are parallel to
; prove that
(i)
(ii)
Answer:
… … … … (i)
… … … … (ii)
Therefore from (i) and (ii) we get . Hence proved.
Question 15: Two equal chords and
of a circle with center
, intersect each other at point
inside the circle. Prove that:
(i)
(ii)
Answer:
Draw and
(Theorem 5: A perpendicular to a chord, from the center of the circle, bisects the chord.)
In
In
(radius of the circle)
Consider
is common
Therefore
(Given)
Since
Therefore
Question 16: In the following figure,
is a square. A circle is drawn with
as the center which meets
at
and
at
. Prove that:
(i)
(ii)
Answer:
(i) In
(radius)
(side of square)
Therefore (S.A.S postulate)
(ii) (radius)
(side of a square)
Therefore
Consider
(side of a square)
Therefore
Question 17: The length of common chord of two intersecting circles is . If the diameter of the circles be
and
, calculate the distance between their center.
Answer:
and
(given)
In
Similarly in
Therefore
Question 18: The line joining the mid points of two chords if a circle passes through the center. Prove that the chords are parallel.
Answer:
(given)
bisects
(given)
Therefore (if line drawn from the center bisects the chord, then is is perpendicular to the chord)
Similarly
Therefore (alternate angles)
Therefore
Question 19: In the following figure, the line
; where
and
are the centers of the circles. Show that:
(i)
(ii)
Answer:
… … … … (i)
… … … … (ii)
(i) – (ii) we get
. Hence proved.
Question 20: and
are two equal chords of a circle with center
which intersect each other at right angle at point
. If
and
, show that
is a square.
Answer:
and
(given)
Since and
and because all angles are
we can say
Therefore is square.
Question 21: In the given figure,
is the center of the circle.
and
are two chords of circle.
and
.
,
. Find the
(i) radius of the circle
(ii) length of chord [2014]
Answer:
,
,
Therefore
(i) Consider
Radius of the circle.
(ii) Now consider