Question

ABC and BDE are two equilateral triangle such that BD =2/3BC.
find the ratio of the area of triangle ABC and BDE

Answer 1

In triangle ABC,
Since it is an equilateral traingle.
          AB =BC = AC
Also in triangle BDE,
 
               BD = DE = BE

Also it is given that,
                     BD = 2/3 BC

Also we know area of equilateral triangle 

     area = (root3/4) * a^2
      
in triangle ABC
          Area                          A1 = root3/4 * BC^2

Also, in triangle BDE,
                    Area                 A2 = root3/4 * BD^2

Ratio of the areas 

                          R = A1/A2
                          R = BC^2/BD^2
      Putting the value of BD
            
R = BC^2/ (2/3BC)^2
            R = 9/4
Thus the ratio of the areas of triangle to area of triangle BDE

R = 9:4

Answer 2

In triangle ABC,

Since it is an equilateral traingle.

         AB =BC = AC

Also in triangle BDE,

 

              BD = DE = BE

Also it is given that,

                    BD = 2/3 BC

Also we know area of equilateral triangle 

    area = (root3/4) * a^2

     

in triangle ABC

         Area                          A1 = root3/4 * BC^2

Also, in triangle BDE,

                   Area                 A2 = root3/4 * BD^2

Ratio of the areas 

                         R = A1/A2

                         R = BC^2/BD^2

     Putting the value of BD

           

R = BC^2/ (2/3BC)^2

           R = 9/4

Thus the ratio of the areas of triangle to area of triangle BDE

R = 9:4

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