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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. www.clix.to/jenardan