Chapter 9 : Circle

Exercise - 19

Q1. ABC is a triangle in which AB = AC. D and E are points on the sides AB and AC respectively such that AD = AE. Show that B, C, E and D are concyclic.

Q2. ABCD is a quadrilateral in which AD = BC and Show that the points A, B, C and D lie on a circle.

Q3. A quadrilateral ABCD is inscribed in a circle so that AB is a diameter of the circle. If

Q4. In the given figure, ABCD is a cyclic quadrilateral, AE || CD is drawn. BA is produced to F. If

Q5. If two sides of a cyclic quadrilateral are parallel, prove that

1. The remaining two sides are equal and
2. Both the diagonal are equal

Q6. In the given figure, O is the centre of the circle. Arc ABC subtends an angle of 130⁰ at the centre O. AB is produced to P. Find

Q7. Prove that any cyclic parallelogram is a rectangle.

Q8. Two opposite sides AB and DC of a cyclic quadrilateral ABCD intersect in E, when produced, Prove that triangles EAD and ECB are similar.

Q9. Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.

Q10. In the adjoining figure. PQTS is a cyclic quadrilateral and O is the centre of the circle. Find .

Q11. Side DC of a cyclic quadrilateral ABCD is produced to a point E. Prove that .

Q12. ABCD is a cyclic quadrilateral with AD|| BC. Prove that AB = DC.

Q13. ABCD is a cyclic quadrilateral whose diagonal AC and BD intersect at P. If AB = DC, prove that

(i)
(ii) PA = PD and PC = PB
(iii) AD || BC

Q14. In the given figure, 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.

Q15. In the given figure, AB is a diameter of the circle with centre O and Chord CD is equal to radius OC. AC and BD produced to meet at P. prove that .

Q16. Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F, respectively. Prove that the angles of the are , respectively.

Q17. The circle passing through the vertices A, B and C of a parallelogram ABCD intersect side CD (or CD produced) at the point P. prove that AP = AD.

Answers