Mathematics: The longer leg of a 30-60-90 triangle is 6. what is the length of the hypotenuse?

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The longer leg of a 30-60-90 triangle is 6. what - id3060906864007, breannacaraballo

jdkrisdaimcc11

12.03.2022

A 30-60-90 triangle is half an equlateral triangle (refer to the image below).

If we call the length of the side $\overline{BD} = x$ , we can see that it is exactly half of the side of the equilateral triangle. So, we have

$\overline{AB} = \overline{AC} = \overline{BC} = 2\cdot \overline{BD} = 2x$

Moreover, we can find the height $\overline{CD}$ using the pythagorean theorem, having

$\overline{CD} = \sqrt{4x^2-x^2} = \sqrt{3x^2} = x\sqrt{3}$ .

Now, you know the longer leg to be 6. The longer leg is the height, so you have

$\overline{CD} = x\sqrt{3} = 6 \implies x = \overline{BD} = \cfrac{6}{\sqrt{3}} = \cfrac{6\sqrt{3}}{3} = 2\sqrt{3}$

So, the hypotenuse is twice that value:

$BC = 2x = 2\cdot 2\sqrt{3} = 4\sqrt{3}$

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