MathematicsState true or false. D(7,9),E(1,1)   and F(−3,−7)   are the vertices of a triangle DEF. L(4,5),M(−1,−3)   and N(2,1)   are the mid-points of DE, EF and FD respectively. Then,  area of ΔDEF  area of ΔLMN = 4 1  .

State true or false.


D(7,9),E(1,1)   and F(3,7)   are the vertices of a triangle DEF. L(4,5),M(1,3)   and N(2,1)   are the mid-points of DE, EF and FD respectively. Then,  area of ΔDEF  area of ΔLMN = 4 1  .


  1. A
    True
  2. B
    False 

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    Solution:

    Given that, D(7,9),E(1,1)   and F(3,7)   are the vertices.
    L(4,5),M(1,3)   and N(2,1)   are the midpoints of DE, EF and FD.
    The area of the triangle with the coordinates ( x 1 , y 1 ),( x 2 , y 2 ),( x 3 , y 3 )   is given by the formula,
    A= 1 2 x 1 y 2 y 3 + x 2 y 3 y 1 + x 3 y 1 y 2  
    Here,
    ( x 1 , y 1 )=(7,9) ( x 2 , y 2 )=(1,1) ( x 3 , y 3 )=(3,7)  
    Area of ΔDEF:  
    1 2 x 1 y 2 y 1 + x 2 y 3 y 1 + x 3 y 1 y 2 1 2 7 19 +1 16 3 91 1 2 561624 1 2 96 =48 square units    Area of ΔLMN:  
    Here,
    ( x 1 , y 1 )=(4,5) ( x 2 , y 2 )=(1,3) ( x 3 , y 3 )=(2,1)  
    12 square units area ofΔDEF area ofΔLMN = 48 12 = 4 1  
    Hence,  area of ΔDEF  area of ΔLMN = 4 1  .
    Hence, the statement is true.
    Therefore, option 1 is correct.
     
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