Maths-
General
Easy

Question

In triangle ABC, medians AD and BE are drawn. If AD=4, straight angle DAB equals pi divided by 6 and straight angle ABE equals pi divided by 3, then the area of the △ABC is

  1. 16 over 3
  2. fraction numerator 32 over denominator 3 square root of 3 end fraction
  3. fraction numerator 64 over denominator 3 square root of 3 end fraction
  4. 8 over 3

hintHint:

Every triangle has 3 medians, one from each vertex. AE, BF and CD are the 3 medians of the triangle ABC. The 3 medians always meet at a single point, no matter what the shape of the triangle is. The point where the 3 medians meet is called the centroid of the triangle.

The correct answer is: fraction numerator 32 over denominator 3 square root of 3 end fraction


    Given; AD=4, and BD=DC
     
    Since the centroid O divides the line AD in the ratio 2:1
     
    AO=8 over 3 space a n d space D O space equals space 4 over 3. 
     
    In ABO,
    tan straight pi over 3 space equals space fraction numerator A O over denominator B O end fraction space rightwards double arrow B O space equals space A O space c o t straight pi over 3
B O space equals space 8 over 3 cross times fraction numerator 1 over denominator square root of 3 end fraction space equals fraction numerator 8 over denominator 3 square root of 3 end fraction
     img
    A r e a space o f space increment A D B space equals space fraction numerator 1 over denominator 2 space end fraction cross times A D cross times B O space space space space space space space space space
space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 space end fraction cross times 4 cross times fraction numerator 8 over denominator 3 square root of 3 end fraction
space space space space space space space space space space space space space space space equals fraction numerator 16 over denominator 3 square root of 3 end fraction
S i n c e comma space M e d i a n space d i v i d e s space a t r i a n g l e space i n space t o space t w o space t r i a n g l e s space o f space e q u a l space a r e a.
A r e a space o f space increment A B C space equals space 2 cross times A r e a space o f space increment A D B
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 space cross times fraction numerator 16 over denominator 3 square root of 3 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 32 over denominator 3 square root of 3 end fraction s q. u n i t s
     

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