MathematicsThe coordinates of A,B,   C  and D   are 6,3 ,   (−3,5),   4,−2  and (x,3x)   respectively. If ar(ΔDBC):ar(ΔABC)=1:2,   then the value of x   from the following choices is:

The coordinates of A,B,   C  and D   are 6,3 ,   (3,5),   4,2  and (x,3x)   respectively. If ar(ΔDBC):ar(ΔABC)=1:2,   then the value of x   from the following choices is:


  1. A
    1.25
  2. B
    2
  3. C
    1.375
  4. D
    2.25 

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    Solution:

    It is given that the coordinates are A(6, 3), B(-3, 5), C(4, -2), D(x, 3x).
    We first find the area of the triangles ABC,DBC   and then find the ratio to equate it to the given ratio 1:2  .
    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  
    Calculation of area of ΔDBC  .
    Here,
    ( x 1 , y 1 )=(x,3x) ( x 2 , y 2 )=(3,5) ( x 3 , y 3 )=(4,2)  
    The area of the ΔDBC   is given by,
    ar(ΔDBC)= 1 2 {x(5(2)3(23x)+4(3x5)} = 1 2 (28x14) =(14x7)squnits   Calculation of area of ΔABC  .
    ( x 1 , y 1 )=(6,3) ( x 2 , y 2 )=(3,5) ( x 3 , y 3 )=(4,2)  
    The area of the ΔABC   is,
    ar(ΔABC)= 1 2 {6×73×(5)+4×(2)} = 1 2 ×49 = 49 2 squnits  
    Hence,
    ar(ΔDBC):ar(ΔABC)=1:2 (14x7) 49 2 = 1 2 28x14= 49 2 56x28=49 x= 77 56 =1.375  
    The value of x   is 1.375  .
    Hence, option 3) is correct.
     
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