Ok...so I know what the 1:2:square root of 3 means and where each of the ratios goes on the 30 and 60 degrees right triangle, but the question is - I don't know how to apply them! It will be great if someone could provide an example for me and explain it, thanks!

]]>The biggest takeaway from these triangle is that the

These are all 30-60-90 triangles! The greatest thing is that you know that whenever you see 30-60-90, the ratios are \(1:\sqrt{3}:2\), but the reverse is also true! Whenever you see that the ratio is \(1:\sqrt{3}:2\), it's a 30-60-90 triangle! ]]>

Basically, whenever you see ANY 30-60-90 triangle, those ratios hold true. 30-60-90 triangles are everywhere.

Here's a good example:

**Question:**

let's say that the purple hexagon has 6 sides all of length 1. what is the perimeter of the blue rectangle?

Try solving this first, and if you're stuck read on to see the solution!

**SOLUTION:**

If you recall, a regular hexagon has angles of \(120^{\circ}\). This means that an exterior angle of the hexagon is \(60^{\circ}\). Then, we can use the 30-60-90 ratios to figure out the legs of the blue triangle, since we know the hypotenuse is equal to \(1\) (it's a side of the hexagon).

You should get these values:

You can do this with all four triangles!

So, the top side turns out to have a total length of \(\frac12+1+\frac12=2\), and the left side turns out to have a total length of \(\frac{\sqrt3}{2}+\frac{\sqrt3}{2}=\sqrt3\). Since it's a rectangle, the total perimeter is just \(2+\sqrt3+2+\sqrt3=\boxed{4+2\sqrt3}\)

I know this is some really new stuff, so if you have any questions, feel free to ask! Also, if there's anything you want me to go more in depth on I can do that too

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