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CIRCLES
CIRCLES
CONCEPT : THEOREMS FROM NCERT
Broad weight age: 6/9 Marks
1.
If two arcs of a circle (or of congruent circles) are congruent, then the
corresponding chords are equal.
2.
If two chords of a circle (or of congruent circles) are equal, then corresponding arcs
(minor, major or semicircular) are congruent.
3.
The perpendicular from the center of a circle to a chord bisects the chord.
4.
The line joining the center of a circle to the mid point of a chord is perpendicular to
the chord.
5.
Equal chords of congruent circles are equidistant from the corresponding centers.
6.
Chords of congruent circles which are equidistant from the corresponding centers,
are equal.
7.
Equal chords of a circle are equidistant from the center.
8.
Chords of a circle which are equidistant from the center are equal.
9.
The angle subtended by an arc of a circle at the center is double the angle subtended
by it at any point on the remaining part of the circle.
10. The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
11. Prove that an angle in a semicircle is a right angle.
12. Prove that the angle in the same segments of a circle is equal.
13. Prove that the arc of a circle subtending a right angle at any point of the circle in its
alternate segment is semicircular.
14. Any angle subtended by a minor arc in the alternate is acute and any angle
subtended by a major arc in the alternate segment is obtuse.
15. If the line segment joining two points subtends equal angles at two other points lying
on the same side of the line segment, the four points are concyclic, i.e., lie on the
same circle.
16. Of any two chords of a circle show that the one which is greater is nearer to the
center.
17. Of any two chords of a circle show that the one which is nearer to the center is
greater.
18. If the two angles of a pair of opposite angles of a quadrilateral are supplementary,
then the quadrilateral is cyclic.
Graphics By:- Roshan Dhawan
- 28 -
Written By:- R. K. Badhan
CIRCLES
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If a side of a cyclic quadrilateral is produced, then the exterior angle is equal to the
interior opposite angles.
CONCEPT : PROBLEMS BASED ON ANGLES
In the figure, points A, B, C and D concyclic and CBE  130 . Find FDC .
In figure AB is the chord of a circle with center O. AB is produced to C such that
BC = OB. OC is joined and produced to meet the circle in D. If ACD  y and
AOD  x prove that x = 3y.
In figure ABC is a triangle is which BAC  30 . Show that BC is the radius of the
circumcircle of  ABC whose center is O.
In the given figure, find x and y.
In the figure ABCD is a cyclic quadrilateral. AE is drawn parallel to CD and BA is
produced. If ABC  92, FAE  20 , find BCD .
In the given figure, O is the center of the circle. The angle subtended by the arc
BCD at the center is 140°. BC is produced to P. Determine BAD and DCP .
In the given figure, A, B, C are three points on a circle such that the angles
subtended by the chords AB and AC at the center O are 80° and 120° respectively.
Determine BAC and the degree measure of arc BPC.
In the given figure, O is the center of the circle and the measure of arc ABC is 100°.
Determine ADC and ABC .
In figure BC = DC and DBC  30 . Find the measure of BAC .
In the figure,  ABC is an isosceles  with AB = AC and mABC  50 . Find
mBDC and mBEC.
In the given figure, C and D are points on the semicircle described on BA as
mBAD  70
mDBC  30. Calculate
mABD
diameter. Given
and
and mBDC .
In the given figure, PQ is a diameter of the circle with center O. Find
QPR, PRS and QPM .
In figure,  ABC is an equilateral  . Find mBEC .
In the given figure, calculate mPQB .
ABCD is cyclic quadrilateral in which BC || AD, ADC  110 and BAC  50 .
Find DAC .
ABCD is a cyclic quadrilateral, if BCD  100 and ABD  70 , find ADB .
In the given figure, calculate mAOC .
In the given figure, ABCD is cyclic quadrilateral; O is the center is the circle. If
BOD  160 , find mBCD and mBPD .
Graphics By:- Roshan Dhawan
- 29 -
Written By:- R. K. Badhan
CIRCLES
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In figure, AOC  130 . Find mABC .
In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P. If
DBC  70 and CAB  30 then find BCD .
In the given figure, O is the center of the circle. If OAB  50 and OCB  60
find AOC .
In the given figure ABC is an equilateral triangle, find BDC and BEC .
In the given figure, O is the center of the circle. If BOD  160 , find x and y.
In figure, PQRS is a cyclic quadrilateral. Find the measure of each of its angles.
In the given figure, if DBC  70 and BAC  40 . Find BCD .
A, B, C, D is the vertices of the cyclic quadrilateral. Find x.
In the given figure, ABCD is a cyclic quadrilateral whose diagonal intersects at P. If
DBC  80 and BAC  40 then find BCD .
In the given figure, ABCD is a cyclic quadrilateral in which AB || DC. If D  70 ,
find the remaining angles.
In the given figure, BDC  35 and ACB  40 . Find ABC .
ABCD is a cyclic quadrilateral is which AD || BC. Prove that B  C .
On a semicircle with AB as diameter a point is taken so that CAB  30 . Find
mABC .
CONCEPT : MISCELLANEOUS PROBLEMS
Two opposite angles of a cyclic quadrilateral are such that one angle is double the
other. Find the measure of the larger angle.
L and M are the mid-points of two equal chords AB and CD and O is the center of
the circle. Prove that: (i) mOLM  mOML, (ii) mALM  mCML .
If two chords are equally inclined to the diameter through point of intersection, they
are equal. Prove.
AB and CD are equal chords of a circle whose center is O. OM  AB and
ON  CD . Prove that OMN  ONM .
In the given figure, O is the center of the circles, BO is the bisector of ABC . Show
that in  ABC , AB = BC.
If one pair of opposite sides of a cyclic quadrilateral are equal, show that its
diagonals are equal.
The bisectors of the opposite angles P and R of cyclic quadrilateral PQRS intersect
the corresponding circle at the points A and B respectively. Prove that AB is a
diameter of the circle.
In the given figure, l is the intersecting the two concentric circles, whose common
center is O, at the points A, B, C and D. Show that : AB = CD.
Graphics By:- Roshan Dhawan
- 30 -
Written By:- R. K. Badhan
CIRCLES
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21.
If two sides of a cyclic quadrilateral are parallel, prove that remaining two sides are
equal.
ABCD is a cyclic quadrilateral. AB and DC when produced meet at E. Prove that
 EBC and  EDA are equiangular.
Two circles intersect each other at P and Q. From P diameters PA and PB are
drawn to two circles. Show that A, Q, B are collinear.
Prove that the circle drawn on any one of the equal sides of an isosceles triangle as
diameter bisects the base.
In figure, O is the center of circle of radius 5 cm. OP  AB , OQ  CD , AB || CD,
AB = 6 cm and CD = 8 cm. Determine PQ.
 ABC is right angled at B. On the side AC, a point, D is taken such that AD = DC
and AB = BD. Find mCAB .
Prove that the quadrilateral formed by the angle bisectors of a cyclic quadrilateral
is also cyclic.
In figure, two circles with center A and B and of radii 5 cm and 3 cm touch each
other internally. If the perpendicular bisectors of segment AB meets the bigger
circle in P and Q, find the length of PQ.
In  PQR , right angled at Q, a point S is taken on the side PR such that PS = SR
and QR = QS. Find mQSR .
ABCD is quadrilateral in which AD = BC and ADC  BCD , show that the points
A, B, C and D lie on a circle.
If two sides of a cyclic quadrilateral are parallel, prove that (i) the remaining two
sides are equal and (ii) both the diagonals are equal.
Prove that any four vertices of a regular pentagon lie on a circle.
Prove that any cyclic parallelogram is a rectangle.
Two chords AB and AC of a circle are equal. Prove that the center of the circle lies
on angle bisectors of BAC .
FIGURES
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Graphics By:- Roshan Dhawan
- 31 -
Written By:- R. K. Badhan