# NCERT Mathematics Class 9 Exemplar Ch 10 Circles Part 7 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Exercise 10.4**

Q. 1. If two equal chords of a circle intersect; prove that the parts of one chord are separately equal to the parts of the other chord.

Q. 2. If non-parallel sides of a trapezium are equal, prove that it is cyclic.

Q. 3. If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.

Q. 4. ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Q. 5. Prove that angle bisector of any angle of a triangle and perpendicular bisector of the opposite side if intersect, they will intersect on the circumcircle of the triangle.

Q. 6. If two chords AB and CD of a circle AYDZBWCX intersect at right angles (see Fig. 10.18) , prove that semicircle.

Q. 7. If ABC is an equilateral triangle inscribed in a circle and P is any point on the minor arc BC which does not coincide with B or C, prove that PA is angle bisector of ∠ BPC.

Q. 8. In Fig. 10.19, AB and CD are two chords of a circle intersecting each other at point E. Prove that

(Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre) .

Q. 9. If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the points P and Q; prove that PQ is a diameter of the circle.

Q. 10. A circle has radius cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in major segment is .

Q. 11. Two equal chords AB and CD of a circle when produced intersect at a point P. Prove that

Q. 12. AB and AC are two chords of a circle of radius r such that . If p and q are the distances of AB and AC from the centre, prove that