## Chapter 11 Circles Ex 11.4

**Question 1.** **Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.** **Solution:**

Let O and O’ be the centres of the circles of radii 5 cm and 3 cm, respectively.

Let AB be their common chord.

**Question 2.** **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.** **Solution:** **Given:** MN and AS are two chords of a circle with centre O, AS and MN intersect at P and MN = AB

**Question 3.** **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 chords.** **Solution:** **Given:** RQ and MN are chords of a with centre O. MN and RQ intersect at P and MN = RQ **To prove:** ∠ OPC = ∠ OPB

**Construction:** Draw OC ⊥ RQ and OB ⊥ MN.

Join OP. **Proof:** In ∆ OCP and ∆ OBP, we get

∠ OCP = ∠ OBP (Each = 90°)

OP = OP (Common)

OC = OB (Equal chords of a circle are equidistant from the centre)

∴ By RHS criterion of congruence, we get

∆ OCP ≅ ∆ OBP

∴ ∠ OPC = ∠ OPB (By CPCT)

**Question 4.** **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).**

**Solution:**

Let OP be the perpendicular from O on line l. Since, the perpendicular from the centre of a circle to a chord

Now, BC is the chord of the smaller circle and OP ⊥ BC.

∴ BP = PC ……(i)

Since, AD is a chord of the larger circle and OP ⊥ AD.

∴ AP = PD …(ii)

On subtracting Eq. (i) from Eq. (ii), we get

AP – BP = PD – PC

⇒ AB = CD

Hence proved.

**Question 5.**
**Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m 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?**
**Solution:**

Let O be the centre of the circle and Reshma, Salma and Mandip are represented by the points Ft, S and M, respectively.

Let RP = xm.

From Eqs. (i) and (ii), we get

Hence, the distance between Reshma and Mandip is 9.6 m.

**Question 6.** **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.** **Solution:**

Let Ankur, Syed and David standing on the point P, Q and R.

Let PQ = QR = PR = x

Therefore, ∆ PQR is an equilateral triangle. Drawn altitudes PC, QD and RN from vertices to the sides of a triangle and intersect these altitudes at the centre of a circle M.

As PQR is an equilateral, therefore these altitudes bisects their sides.

In ∆ PQC,

PQ^{2} = PC^{2} + QC^{2} (By Pythagoras theorem)