# NCERT Mathematics Class 9 Exemplar Ch 10 Circles Part 5

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**Exercise 10.3**

Q.1. If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

Answer:

Q.2. If the perpendicular bisector of a chord AB of a circle PXAQBY intersects the circle at P and Q, prove that arc

Q.3. A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC and CA are concurrent.

Q.4. AB and AC are two equal chords of a circle. Prove that the bisector of the angle BAC passes through the centre of the circle.

Q.5. If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

Q.6. ABCD is such a quadrilateral that A is the centre of the circle passing through B, C and D. Prove that

Q.7. O is the circumcenter of the triangle ABC and D is the mid-point of the base BC. Prove that.

Q.8. On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that

Q.9. Two chords AB and AC of a circle subtend angles equal to 90º and 150º, respectively at the centre. Find ∠ BAC, if AB and AC lie on the opposite sides of the centre.

Answer:

Q.10. If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, prove that the points B, C, M and N are concyclic.

Q.11. If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove that the quadrilateral so formed is cyclic.

Q.12. If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.

Q.13. The circumcenter of the triangle ABC is O. Prove that

Q.14. A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in major segment.

Answer:

Q.15. In Fig.10.13, and chord Find

Answer: