What is a 30-60-90 triangle? Please give an example.

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What is a 30-60-90 triangle? Please give an example.

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Answer 1 · 2015-11-13T22:23:50

A 30-60-90 triangle is a right triangle with angles $30^{\circ}$ $30^@$ , $60^{\circ}$ $60^@$ , and $90^{\circ}$ $90^@$ and which has the useful property of having easily calculable side lengths without use of trigonometric functions.

Explanation:

A 30-60-90 triangle is a special right triangle, so named for the measure of its angles. Its side lengths may be derived in the following manner.

Begin with an equilateral triangle of side length $x$ $x$ and bisect it into two equal right triangles. As the base is bisected into two equal line segments, and each angle of an equilateral triangle is $60^{\circ}$ $60^@$ , we end up with the following
enter image source here
Because the sum of the angles of a triangle is $180^{\circ}$ $180^@$ we know that $a = 180^{\circ} - 90^{\circ} - 60^{\circ} = 30^{\circ}$ $a = 180^@ - 90^@ - 60^@ = 30^@$

Furthermore, by the Pythagorean theorem, we know that
${(\frac{x}{2})}^{2} + h^{2} = x^{2}$ $(x/2)^2 + h^2 = x^2$
$\Rightarrow h^{2} = \frac{3}{4} x^{2}$ $=>h^2 = 3/4x^2$
$\Rightarrow h = \frac{\sqrt{3}}{2} x$ $=>h = sqrt(3)/2x$

Therefore a 30-60-90 triangle with hypotenuse $x$ $x$ will look like
enter image source here

For example, if $x = 2$ $x = 2$ , the side lengths of the triangle will be $1$ $1$ , $2$ $2$ , and $\sqrt{3}$ $sqrt(3)$

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What is a 30-60-90 triangle? Please give an example.

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