# Circles

**Exercise 1**

**1. ** Fill in the blanks:

(i) The centre of a circle lies in of the circle.

(ii) A point, whose distance from the centre of a circle is greater than its radius lies in of the circle.

(iii) The longest chord of a circle is a of the circle.

(iv) An arc is a when its ends are the ends of a diameter.

(v) Segment of a circle is the region between an arc and of the circle.

(vi) A circle divides the plane, on which it lies, in parts.

**2. **Write True or False:

(i) Line segment joining the centre to any point on the circle is a radius of the circle.

(ii) A circle has only finite number of equal chords.

(iii) If a circle is divided into three equal arcs each is a major arc.

(iv) A chord, which is twice as long as its radius is a diameter of the circle.

(v) Sector is the region between the chord and its corresponding arc.

(vi) A circle is a plane figure.

**Exercise 2**

**3. **Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

**4. ** Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

**Exercise 3**

**5. **Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

**6. **Suppose you are given a circle. Give a construction to find its centre.

**7. ** If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

**Exercise 4**

**8. ** Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centers is 4 cm. Find the length of the common chord.

**9. **If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

**10. **If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chord.

**11. **If a line intersects two concentric circles (circles with the same centre) with centre 0 at A, B, C and D, prove that AB = CD. (See figure)

**12.**Three girls Reshma, Salma and Mandip are standing on a circle of radius 5 m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6 m each, what is the distance between Reshma and Mandip?

**13. **A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

**Exercise 5**

**14.** In figure, A, B, C are three points on a circle with centre 0 such that ∠ BOC = 30°, ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ ADC.

**15. ** A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord on a point on the minor arc and also at a point on the major arc.

**16. **In figure, ∠ PQR = 100°, where P, Q, R are points on a circle with centre O. Find ∠ OPR.

**17. ** In figure, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

**18. ** In figure, A, B, C, D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.

**19.**ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. ∠ DBC = 70°, ∠ BAC is 30° find ∠ BCD. Further if AB = BC, find ∠ ECD.

**20.** If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

**21.** If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

**22. ** Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D, P, Q respectively (see figure). Prove that ∠ ACP = ∠ QCD.

**23.** If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

**24.** ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ ABD.

**Exercise 6**

**25.** Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

**26.** Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find he radius of the circle.

**27.** The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord form the centre?

**28.** Let vertex of an angle ABC be located outside a circle and let the sides of the angle intersect chords AD and CE with the circle. Prove that ∠ ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.

**29.** Prove that the circle drawn with any drawn with any side of a rhombus as a diameter, passes through the point of intersection of its diagonals.

**30. ** ABCD is a parallelogram. The circle through A, B and C intersect CD (produced it necessary) at E. Prove that AE = AD.

**31. ** AC and BD are chords of a circle which bisect each other. Prove that:

(i) AC and BD are diameters.

(ii) ABCD is a rectangle.

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

**33.** Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.