Floating

Geometry Level 4

What is the maximum area of the part colored blue?


The answer is 1.25.

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1 solution

It is an isosceles triangle with vertex angle 60 degrees.
So it is an equilateral triangle.
So its area is 3 4 ( 7 4 5 ) 2 = 15 4 . S i n c e t h e t o p v e r t e x a n g l e i s h a l f t h a t o f t h e o n e a t t h e m i d d l e b y t h e s a m e h o r i z o n t a l s i d e , t h e m i d d l e p o i n t i s t h e c i r c u m c e n t e r . t h e b l u e a r e a = 1 3 t r i a n g l e a r e a . b l u e a r e a = 1 3 15 4 = 1.25. \dfrac{\sqrt3} 4*(\sqrt[4]75)^2=\dfrac{15}4. \\ Since~the~top~vertex~angle~is~half~that~ of~ the ~one~at ~the~middle~by~the~same~horizontal~ side,\\ the ~middle~ point~ is~ the~ circumcenter.\\ \therefore~the~blue~area~=~\frac 1 3~ triangle~area.\\ \therefore~blue~area~=~\frac 1 3 * \dfrac{15} 4=\Large~~~\color{#D61F06}{1.25}.

I got a wrong answer because the quality of the picture is so bad I did not read the side length was sqrt(sqrt(75)). This is really disappointing...

Gerard Boileau - 2 years, 7 months ago

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