Download CIRCLES class X

THE INDIAN COMMUNITY SCHOOL, KHAITAN
MATHEMATICS
CIRCLES
class X
Assignment # 9
1. Prove that the line joining the mid-point of a chord to the centre of a circle
passes through the mid-point of the corresponding minor arc.
2. If a pair of opposite sides of a cyclic quadrilateral is equal, prove that the
other two sides are parallel.
3. ABCD is a cyclic quadrilateral. A circle passing through A and B meets AD
and BC in the points E and F respectively. Prove that EF  DC
4. A hexagon ABCDEF is inscribed in a circle. Prove that sum of its alternate
angles  B,  D,  F is equal to four right angles.
5. Prove that sum of the angles in the four segments exterior to a cycle
quadrilateral is equal to 6 right angles.
6. A chord of a circle is equal to its radius. Prove that the angle subtended by
the chord a point on the circumference of the circle is either 30 0 or 1500
7. Prove that every cyclic parallelogram is a rectangle.
8. Prove that any four vertices of a regular pentagon are concyclic.
9. Prove that the circle drawn with any side of a rhombus as diameter passes
through the point of intersection of diagonals.
10. The diagonals of a cyclic quadrilateral are at right angles. Prove that the
perpendicular from the point of intersection on any side when produced
backward bisects the opposite side.
11. PQRS is a cyclic quadrilateral whose diagonals intersect at right angles at T.
Prove that any line passing through T and bisecting any side of the
quadrilateral is perpendicular to the opposite side.
12. PQR is an equilateral triangle inscribed in a circle. S is any point on the minor
arc QR. Prove that PS= QS + SR.
13. ABCD is a quadrilateral in which AB=AD=AC. Prove  BAD =2(  CBD+  CDB).
14. In a circle with center O, chords PQ and RS intersect at A. Prove  POR
+  QOS = 2  PAR
15. The bisectors of the opposite angles P and R of a cyclic quadrilateral PQRS
intersect the corresponding circle at the points A and B respectively. Prove
that AB is a diameter of the circle.
16. AB and CD are two equal chords of a circle whose center is O. When
produced these chords meet at E. Prove that EB = ED and EA = EC.
17. O is the centre of the circle and P, Q, and R are three points on the minor
arc. Prove that  POR = 2(  PRQ +  QPR
18. A line ‘l’ is intersecting the two concentric circles (O, r) at the point A, B, C,
and D. Show that AB =CD
19. If two circles intersect in two points, prove that the line through the
centers is the perpendicular bisector of the common chord.
20. Of any two chords of a circle, prove that the one, which is longer, is nearer
to the centre.
21. Prove that the quadrilateral formed the angle bisectors of a cyclic
quadrilateral is also cyclic.
22. K, L, and M and N are respectively the midpoints of equal chords of AB, CD,
EF, and GH of a circle with centre O. Prove that K.L, M and N line on a circle
with centre O.
23. Two circles with centers A and B of radii 5 cm and 3 cm touch each other
internally. If the perpendicular bisector of segment AB meets the bigger
circle in P and Q, find the length of PQ.
24. ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle. If
 ADC = 1300. Determine  ABC and  BAC.
25. Two circles C (0,r) and C (0’,r’) intersect P and Q. A line through P parallel to
OO’ intersects the circle at A and B. Prove that AB = 2 OO’.
26. If a pair of opposite sides of a cyclic quadrilateral is parallel, prove that the
other two sides are equal.
27.
O is the centre of the circle with radius 5 cm. OP  AB, OQ  CD, AB  CD ,
AB=8 cm, CD = 6 cm. Determine PQ.
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