Question relates to the ratio 1:2:square root of 3 in a 30 and 60 right triangle
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Hey, ya'all!
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!
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@victorioussheep These ratios are actually crucial! They're very common in a ton of competition math problems, but most importantly, those ratios are fundamental when dealing with trigonometry later on. But no need to worry about that yet!
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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@quacker88 I think I know what's going on now... the thing that puzzled my mind when I first heard about the 30-60-90 triangle is I don't know how to plug in the number. Sometimes my brain needs to take a big circle to understand what is really going on.... anyway, thank you for explaining this with a sample question! btw I also learned about the 45-45-90 triangle today...ah...
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@victorioussheep Yes the 45-45-90 triangle is very important too!
The biggest takeaway from these triangle is that the RATIOS are always the same. In every 30-60-90 triangle, the ratio of the hypotenuse to the short leg (the side opposite 30 degree) is ALWAYS 2:1.
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!
