Thursday, 12 January 2012

SOME IMPORTANT QUESTIONS (Chapter:Circles)


Q1. ABC is an isosceles triangle in which AB = AC, circumscribed about a circle. Prove that the base is bisected by the point of contact. i.e. if AB = AC, prove that BE = EC.                 CBSE 2008
Solution:-
Since the tangents drawn from any exterior point to a circle are equal in length, therefore:-
AD = AF - ( Tangents from A)
BD = BE - ( Tangents from B)
CE = CF - ( Tangents from C)
Now,
AB = AC
AB- AD = AC- AD - (Subtracting AD from both sides)
AB- AD = AC – AF - (Since, AD=AF)
BD = CF
BE = CF
BE = CE , Hence proved.




Q2. A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.
                                                                                                                               CBSE 2008,2009
Solution:-

Since the tangents drawn from any exterior point to a circle are equal in length, therefore:-
AP = AS - (Tangents from A) - (i)
BP = BQ - ( Tangents from B) - (ii)
CR = CQ - ( Tangents from C) - (iii)
DR = CQ - ( Tangents from D) - (iv)

Adding (i) , (ii) , (iii) , (iv) , we get :-
AP + BP + CR + DR = AS + BQ + CQ + DS
(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)
AB + CD = AD + BC
AB + CD = BC + DA , Hence proved.






Q3. Prove that a parallelogram circumscribing a circle ia a rhombus.                   CBSE 2002, 2008
Solution:-
Let ABCD be a parallelogram such that its sides touch a circle with centre O.
Since the tangents to a circle from an exterior point are equal in length , therefore:-
AP = AS - (Tangents from A) ….......(i)
BP = BQ - (Tangents from B) …........(ii).
CR = CQ - (Tangents from C) ….........(iii)
DR = DS - (Tangents from D) …...........(iv).

Adding (i) , (ii) , (iii) , (iv) , we get:-
AP + BP + CR + DR = AS + BQ + CQ + DS
(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)
AB + CD = AD + BC
2AB = 2BC ….....................(Since, ABCD is a parallelogram, therefore AB = CD and BC = AD)
AB = BC
AB = BC = CD = AD
Therefore, ABCD is a rhombus, Hence proved.

Q4. A circle is inscribed in a triangle ABC having sides 8cm , 10cm and 12cm. Find AD , BE and CF.                                                                                                                                 CBSE 2001
Solution:-

Since, the tangents drawn from an external point to a circle are equal , therefore:-
AD = AF = say , x
BD = BE = say , y
CE = CF = say , z
Now,
AB = 12cm , BC = 8cm and CA = 10cm
Therefore,
x + y = 12 , y + z = 8 and z + x = 10
(x+y)+(y+z)+(z+x) = 12 + 8 + 10
2(x+y+z) = 30
x+y+z = 15

Now,
x+y = 12 and x+y+z = 15
12 + z = 15
z = 15 – 12 = 3

Similarly,
y+z = 8 and x+y+z = 15
x+8 = 15
x = 15 – 8 = 7
Similarly,
z+x = 10 and x+y+z = 15
y+10 = 15
y = 15 – 10 = 5
Therefore, x = 7cm , y = 5cm ad z = 3cm i.e. AD = 7cm , BE = 5cm and CF = 3cm.

Q5. Prove that the tangents at the extremities of any chord make equal angles with the chord.
                                                                                                                             CBSE 2000,2001,2002
Solution:-
Let AB be a chord of a circle with centre O , and let AP and BP be the tangents at A and B.
Suppose the tangents meet at P. Join OP. Suppose OP meets AB at C .
We need to prove that PAC = PBC.

In triangles PCA and PCB , we have:-
PA = PB ….......(Since, tangents from an external points are equal)
  APC = BPC ….........(Since, PA and PB are equally inclined to OP)
PC = PC ….......(Common)

therefore, by SAS criterion of congruence , we have:-
triangle PAC = triangle PBC
PAC = PBC , Hence proved.

                                                       More important questions will be posted soon..!!
                                                    Feel free to discuss your problems related any question with us..!!
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3 comments:

  1. problem in last question...........

    ReplyDelete
  2. @gurvinder
    make the diagram of 5th question yourself and give this question a try...still if u found any problem in solving the ques then do tell us..!!!

    ReplyDelete
  3. done! actually i was confused with diagram that was given

    ReplyDelete