__More Special Right Triangles__

<== an equilateral triangle

<== an equilateral triangle
split into two equal triangles. These are called **30˚-60˚
right triangles**. Why? Well,
an equilateral triangle has all angles equal to 60˚. When we put the
height in, it split the top angle in half, into two 30˚ angles. So, in one
of the right triangles, there is a 90˚ angle, a 60˚ angle, and a 30˚ angle.

Now let's say we want to find the height of the equilateral triangle (the side b of the small right triangle). First use the Pythagorean Theorem.

a^{2} + b^{2} = c^{2
}b^{2} = c^{2} - a^{2}

b^{2 =} 2^{2} - 1^{2
}b^{2} = 4 - 1

b^{2} = 3

b = **√3** or **1√3**

Let's try it again if the side
length of the **equilateral** triangle is 4.

b^{2} = c^{2} - a^{2}

b^{2} = 4^{2} - 2^{2
}b^{2} = 16 - 4

b^{2 }= 12

b = √12 = **2√3**

Notice a pattern? :) ... if you don't, the rule is below:

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In a 30˚-60˚ right triangle, if the smallest side is a units, then the hypotenuse will be 2a units, and the longer side will be a√3 units.

Similarly, if the side length of an equilateral triangle is a units, then the height will be ½a√3 units.

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