Three right angles
Work your way through these right-angled triangles to find $x$.
Age
14 to 16
Challenge level
Problem
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
Student Solutions
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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 $
$\Rightarrow x=\dfrac2{cos{30}}=\dfrac 4{\sqrt{3}}$