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Try this beautiful problem on area of Hexagon from Geometry.

Equiangular hexagon \(ABCDEF\) has side lengths \(AB=CD=EF=1\) and \(BC=DE=FA=r\). The area of \(\triangle ACE\) is \(70\%\) of the area of the hexagon. What is the sum of all possible values of \(r\)?

- \(6\)
- \(\frac{2}{\sqrt5} \)
- \(8\)
- \(9\)
- \(9\)

Geometry

Hexagon

Triangle

But try the problem first...

Answer: \(6\)

Source

Suggested Reading

AMC-10A (2010) Problem 19

Pre College Mathematics

First hint

Given that area of \(\triangle ACE\) is \(70\)% of the area of the hexagon.so at first we have to find out the area of the \(\triangle ACE\).Clearly \(\triangle ACE\) is an equilateral triangle.Now from the cosines law we can say that \(AC^2=r^2+1^2-2r cos \frac{2\pi}{3}=r^2+r+1\).Therefore area of \(\triangle ACE=\frac{\sqrt 3}{4}(r^2+r+1)\).Can you find out area of the Hexagon \(ABCDEF\)?

Can you now finish the problem ..........

Second Hint

**Area Of The Hexagon \(ABCDEF\) :**

If we extend $BC$, $DE$ and $FA$ so that $FA$ and $BC$ meet at $X$, $BC$ and $DE$ meet at $Y$, and $DE$ and $FA$ meet at $Z$,Now we can find out area of the hexagon $ABCDEF$ which is formed by taking equilateral triangle $XYZ$ of side length $r+2$. So if we remove three equilateral triangles, $ABX$, $CDY$ and $EFZ$, of side length $1$.

Therefore The area of $ABCDEF$ is \(\frac{\sqrt 3}{4} (r+2)^2 - \frac{3\sqrt 3}{4}=\frac{\sqrt 3}{4} (r^2+4r+1)\)

can you finish the problem........

Final Step

Given that area of \(\triangle ACE\) is \(70\)% of the area of the hexagon.Therefore

\(\frac{\sqrt 3}{4}(r^2+r+1)\)=\(\frac{7}{10}.\frac{\sqrt 3}{4}(r^2+4r+1)\)

\(\Rightarrow r^2-6r+1=0\).Now from Vieta's Relation the sum of the possible value of \(r\) is \(6\)

- https://www.cheenta.com/quadratic-equation-problem-amc-10a-2002-problem-12/
- https://www.youtube.com/watch?v=pYSIvF7jZy4

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