Note: In almost all the problems below the implementation of the following theorem is important. Theorem 3: The perpendicular from the center of a circle to a chord bisects the chord.
Question 1: The radius of the circle is and the length of one of its chords is
. Find the distance of the chord from the center.
Answer:
Refer to the adjoining diagram.
cm
Question 2: The radius of the circle is and the length of one of its chords is
. Find the distance of the chord from the center.
Answer:
Refer to the adjoining diagram.
cm
Question 3: Find the length of a chord which is at a distance of from the center of the circle of radius
.
Answer:
Refer to the adjoining diagram.
cm
Therefore cm
Question 4: Find the length of a chord which is at a distance of from the center of the circle of radius
.
Answer:
Refer to the adjoining diagram.
cm
Therefore cm
Question 5: Find the length of a chord which is at a distance of from the center of the circle of radius
.
Answer:
Refer to the adjoining diagram.
cm
Therefore cm
Question 6: Two chords and
of length
and
respectively of a circle are parallel. If the distance between
and
is
, find the radius of the circle.
Answer:
Refer to the adjoining diagram.
Squaring both sides we get
Squaring both sides we get
Question 7: An equilateral triangle of side is inscribed in a circle. Find the radius of the circle.
Answer:
Refer to the adjoining diagram.
cm
Question 8: is a diameter of the circle.
is a point in
such that
and
. Find the length of the shortest chord through
.
Answer:
Refer to the adjoining diagram.
cm
cm
Therefore
Therefore cm
Hence cm
Question 9: The length of the common chord of two intersecting circles is . If the radii of the two circles are
and
, find the distance between their centers.
Answer:
Refer to the adjoining diagram.
Therefore
cm
Similarly,
Therefore
cm
Hence
cm
Question 10: A rectangle with a side of length is inscribed in a circle of diameter
. Find the area of the rectangle.
Answer:
Refer to the adjoining diagram.
cm
cm
Hence the area of rectangle
Question 11: The center of a circle of radius units is the point
.
is a point inside the circle.
is a chord of the circle such that
. Calculate the length of
.
Answer:
Refer to the adjoining diagram.
cm
Therefore cm
Hence cm
Question 12: and
are two parallel chords of a circle whose center is
and radius is
. If
is
and
is
, find the distance between
and
, if they lie i) on the same side of center
ii) on the opposite side of center
Answer:
Refer to the adjoining diagram.
i) cm
cm
Therefore cm
ii) cm
cm
Therefore cm
Question 13: and
are two parallel chords of a circle such that
and
. If the chords are on the opposite sides of the center and the distance between then is
, find the radius of the circle.
Answer:
Refer to the adjoining diagram.
Similarly,
Therefore
Squaring both sides
Squaring both sides
cm
Question 14: and
are two chords of a circle such that
and
.
. If the distance between
and
is
, find the radius of the circle.
Answer:
Refer to the adjoining diagram.
Squaring both sides
Squaring both sides
cm
Question 15: is an isosceles triangle inscribed in a circle. If
and
, find the radius of the circle.
Answer:
Refer to the adjoining diagram.
cm
Squaring both sides
cm
Question 16: In a circle of radius ,
and
are two chords such that
. Find the length of the chord
.
Answer:
Refer to the adjoining diagram.
Therefore
Squaring both sides
cm
Question 17: Two concentric circles with center have
as the points of intersection with the line
as shown in the diagram. If
and
, find the lengths of
and
.
Answer:
Refer to the adjoining diagram.
cm
cm
cm
cm
cm
cm
Question 18: Two circles of radii and
intersect and the length of the common chord is
. Find the distance between their centers.
Answer:
Refer to the adjoining diagram.
Similarly,
Therefore
cm
Question 19: In the figure, two circles with center and
and of radii
and
touch each other internally. If the perpendicular bisector of segment
meets the bigger circle in
and
, find the length of
.
Answer:
Refer to the adjoining diagram.
Therefore cm