The square

Geometry Level pending

The figure shows a square ABCD , the point F divides the length of DC such that DF : FC = ( n - 1 ) : ( n + 1 ) , where as the point E divides the length of AB such that AE : EB = ( n+ 1 ) : ( n - 1 ) . If the percentage of the shaded area to the area of the square ABCD = 24 % . find the value of n ?


The answer is 5.

Aziz Alasha by Aziz Alasha , 59, Sudan

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1 solution

Tom Engelsman
May 13, 2021

If we place square A B C D ABCD of side length 2 n 2n in the first quadrant of the x y − xy- plane such that:

1) The green parallelogram's other vertices are E ′ E' and F ′ ; F';

2) A = ( 0 , 0 ) A = (0,0) ;

3) △ A D F = △ B C E = 1 2 ( 2 n ) ( n − 1 ) ; \triangle{ADF} = \triangle{BCE} = \frac{1}{2}(2n)(n-1);

4) △ A F ′ E = △ C F E ′ = 1 2 ( n + 1 ) ( n + 1 ) ; \triangle{AF'E} = \triangle{CFE'} = \frac{1}{2}(n+1)(n+1);

then we have:

( 2 n ) 2 − [ 2 ( 1 / 2 ) ( 2 n ) ( n − 1 ) + 2 ( 1 / 2 ) ( n + 1 ) 2 ] ( 2 n ) 2 = 0.24 ; \Large \frac{(2n)^2 - [2(1/2)(2n)(n-1) + 2(1/2)(n+1)^{2}]}{(2n)^2} = 0.24;

or 4 n 2 − ( 3 n 2 + 1 ) 4 n 2 = 0.24 ; \Large \frac{4n^2 - (3n^2+1)}{4n^2} = 0.24;

or ( 1 − 0.96 ) n 2 = 1 ; (1-0.96)n^2 = 1;

or n = 5 . \boxed{n=5}.

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