Geometry

Geometry Level pending

ADGJ is a square.8 points are taken on the four sides of the square as the diagram below:

Here AB=BC=CD=DE=EF=FG=GH=HI=IJ=JK=KL=LA.The area of the black portion is 147 square unit.What is the area of the square ADGJ?


The answer is 189.

Akash Hossain by Akash Hossain , 18, Bangladesh

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1 solution

A r e a [ A D G J ] = ( A D ) 2 Area[ADGJ]=(AD)^2 but A D = A B + A B + A B = 3 ( A B ) AD=AB+AB+AB=3(AB)

A r e a [ s h a d e d ] = ( A D ) 2 − 4 ( 1 2 ) ( A B ) 2 = ( A D ) 2 − 2 ( A B ) 2 Area[shaded]=(AD)^2-4\left(\dfrac{1}{2}\right)(AB)^2=(AD)^2-2(AB)^2

Substitute:

147 = ( 3 A B ) 2 − 2 ( A B ) 2 147=(3AB)^2-2(AB)^2

147 = 9 ( A B ) 2 − 2 ( A B ) 2 147=9(AB)^2-2(AB)^2

147 = 7 ( A B ) 2 147=7(AB)^2

( A B ) 2 = 21 (AB)^2=21

A B = 21 AB=\sqrt{21}

Therefore,

A r e a [ A D G J ] = [ 3 ( A B ) ] 2 = [ 3 21 ] 2 = 189 Area[ADGJ]=[3(AB)]^2=[3\sqrt{21}]^2=\boxed{189}

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