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# Three Right Angles

In the diagram, the green triangle and the blue triangle are congruent, since they are both right-angled triangles with a 30$^\circ$ angle, and they share the side adjacent to the 30$^\circ$ angle.

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.

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:

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

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

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In the diagram, the green triangle and the blue triangle are congruent, since they are both right-angled triangles with a 30$^\circ$ angle, and they share the side adjacent to the 30$^\circ$ angle.

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.

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:

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

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

You can find more short problems, arranged by curriculum topic, in our short problems collection.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.