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30-60-90 Triangles - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A 30 60 90 triangle is a special type of right triangle. What is special about 30 60 90 triangles is that the sides of the 30 60 90 triangle always have the same ratio. Therefore, if we are given one side we are able to easily find the other sides using the ratio of 1:2:square root of three. This special type of right triangle is similar to the 45 45 90 triangle.

There are two special array triangles, 45, 45,
90 and 30, 60, 90 triangles.

Now what's special about this 30, 60, 90.
triangle is that of the three sides that.
are all going to be different, I only
need to know one, and I can find the other two.
So that's pretty amazing, that of the three
sides of a triangle, when it's a
30-degree, a 60-degree and a 90-degree,
I only need to know one side and
I can find the other ones.

So the shortcut, and again this is based
on the Pythagorean theorem, is to find
your longer leg
Now how do I know that. this is my longer leg.
Well, if I look at my angle measures, 30
degrees is my smallest measure, which
means the side that is opposite
of 30 will be my shortest leg.
The next longest will be
my next largest angle.
And since 60 degrees is smaller than 90,
60 degrees is my next longest leg.
And then, of course, our hypotenuse is always
the longest side in our triangle.

So to go from our shorter leg to our longer
leg, I need to multiply by the square
root of 3. So if this side is, let's
say, five units, then your
longer leg will be 5 times the square
root of 3.

Now, let's say you want to find your hypotenuse.
To go from your shorter leg to your hypotenuse,
all you need to do is multiply.
by 2. So this is going to be 2.
times S. So if S is 5, your
hypotenuse will be 10.
So this works in both directions.

So let's say you had your longer leg and
you wanted to find the other two sides.
Well, first -- and actually I forgot to
write down right here that you would
multiply by the square root of 3.
So let's say you wanted to go
in the opposite direction.
From your longer leg to your shorter leg.
Well, the inverse operation of multiplying
by 3 is to divide by the square root.
of 3. The inverse operation of going
from your hypotenuse to your
shorter leg is going to be not multiplying.
by 2 but dividing by 2. So.
so the key leg in this triangle
is this shorter leg.
Because from here you can find your hypotenuse
by multiplying by 2 and you can
find your longer leg.

So use these shortcuts whenever you see.
a triangle with 30, 60 and 90 degrees.

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