## Circles

**Chapter 10**

**Exercise 10.6**

**EX 10.6 QUESTION 1.**

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

**Solution:**

Given: Two circles with centres P and Q respectively such that they intersect each other at P and Q.

To Prove: ∠PAQ = ∠PBQ.

Proof: In ∆APQ and ∆BPQ,

PA = PB [Radii of the same circle]

QA = QB [Radii of the same circle]

PQ = PQ [Common]

∴ ∆APQ = ∆BPQ [ SSS congruence criteria]

⇒ ∠PAQ = ∠PBQ [C.P.C.T.]

**EX 10.6 QUESTION 2.**

**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 , find the radius of the circle.**

**Solution:**

**EX 10.6 QUESTION 3.**

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

**Solution:**

**EX 10.6 QUESTION 4.**

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

**Solution:**

**EX 10.6 QUESTION 5.**

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

**Solution:**

**EX 10.6 QUESTION 6.**

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

**Solution:**

**EX 10.6 QUESTION7.**

**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.**

**Solution:**

**EX 10.6 QUESTION 8.**

**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 triangle DEF are 90°–(½)A, 90°–(½)B and 90°–(½)C.**

**Solution:**

**EX 10.6 QUESTION 9.**

**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.**

**Solution:**

**EX 10.6 QUESTION 10.**

**In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.**

**Solution:**