MathematicsChoose the correct answer:In the figure, CD ¯ , AE ¯ , BF ¯   are one third of their respective sides. It follows that A N 2 ¯ : N 2 N 1 ¯ : N 1 D ¯ =3:3:1   and similarly for lines BE and CF. Then the area of triangle N 1 N 2 N 3   is:

Choose the correct answer:


In the figure, CD ¯ , AE ¯ , BF ¯   are one third of their respective sides. It follows that A N 2 ¯ : N 2 N 1 ¯ : N 1 D ¯ =3:3:1   and similarly for lines BE and CF. Then the area of triangle N 1 N 2 N 3   is:


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  1. A
    1 10 ΔABC  
  2. B
    1 9 ΔABC  
  3. C
    1 7 ΔABC  
  4. D
    1 6 ΔABC   

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    Solution:

    The correct answer is option 3.
    Since CD ¯ , AE ¯  and  BF ¯   lie on  BC, AC and AB respectively
    CD= 1 3 BC 3CD=BC CD:BC=1:2  
    Similarly,
    AE= 1 3 AC AE:AC=1:2 and BF= 1 3 AB BF:AB=1:2  
    Now,
    Area of triangle= 1 2 ×base×height  
    AE= 1 3 AC  
    ΔABE= 1 3 arΔABC  
    Accordingly, A N 2 ¯ : N 2 N 1 ¯ : N 1 D ¯ =3:3:1  
    For lines BE and CF,
    arΔAB N 2 = 6 7 arΔABE  
    From the equations,
    arΔAB N 2 = 6 7 × 1 3 arΔABC arΔAB N 2 = 2 7 arΔABC  
    And
    arΔBFC= 1 3 arΔABC and arΔB N 3 C= 6 7 arΔBFC  
    From above two equations,
    arΔB N 3 C= 1 6 × 1 3 arΔABC arΔB N 3 C= 2 7 arΔABC arΔA N 1 C= 2 7 arΔABC  
    To find the area of triangle N 1 N 2 N 3  
    arΔ N 1 N 2 N 3 =arΔABCarΔAB N 2 arΔB N 3 CarΔA N 1 C  
    But
    The correct answer is option 3.
     
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