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,
PQ2 = PC2 + QC2 (By Pythagoras theorem)
