# Geometry Puzzle

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AKG
Homework Helper
$\sqrt{2.3925} - 1.35 \approx 0.197$

Yes, that appears to be the correct answer. I am interested how you got it. The expression I got was
sin((60-2*(90-arccos(.45)))/2) * 3^.5 * 2

AKG
Homework Helper
2(sin(120)/sin(30))sin((60 - Y)/2)
= 2(sin(120)/sin(30))(sin(30)cos(Y/2) - cos(30)sin(Y/2))
= 2(30.5)(0.5(1 - 0.45²)0.5 - 0.5(30.5)(0.45))
= (3(1 - 0.45²))0.5 - 1.35
= (2.3925)0.5 - 1.35

Y is the small angle in the triangle the contains the blue line (the bottom angle). I didn't want to use any inverse trigonometric functions, so I used Pythogras' Theorem, Sine Law, and the angle addition formula for sine.

NateTG
Homework Helper
Hmm, if you bisect the figure, then you have an isoscoles trianlge at the top so the interior angle of the top triangle is
$$2\sin^{-1}(.45)$$
Moreover, it's not hard to see that the lenght from the center intersection to the edge of the red segment has length
$$\sqrt{3}$$

That means that you can go with:
$$2\left(\sqrt{3} \sin\left(30-\sin^{-1}(.45)\right)\right)$$

Which can easily be simplified using angle addition forumlas.