# Three right angles

Work your way through these right-angled triangles to find $x$.

Below are three right triangles all sharing vertex A. Find the length $x$.

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*This problem is adapted from the World Mathematics Championships*

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The purple triangle is also a right-angled triangle with a 30$^\circ$ angle, but it is not congruent to the others, because the side it shares with the blue triangle is the hypotenuse of the blue triangle, but not of the purple triangle.

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The green and blue triangles make an equilateral triangle:

This means that the side which the pink triangle shares with the blue triangle is 2 cm.

Then $x$ can be found using trigonometry:

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$\cos{30} = \dfrac 2x $

$\Rightarrow x=\dfrac2{cos{30}}=\dfrac 4{\sqrt{3}}$