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Geometry

Geometry: Special Right Triangles

Right triangles & trigonometry

Special right triangles

Special Right Triangles Theorem 2-4-1 (45-45-90 Triangle)
In a 45° - 45°-90° triangle, both legs are congruent and the length of the
hypotenuse is √2 times the length of the leg.
Notice that in this triangle the legs are congruent and the acute angles are
congruent. Sometimes we will see a problem that simply says that either the 2
legs are congruent or the two acute angles are congruent and you will be
expected to know that it is a 45-45-90 triangle.
S
45°
S√2
S
45° Theorem 2-4-2 (30-60-90 Triangle)
In a 30° -60°-90° triangle, the length of the hypotenuse is double the length of
the shorter leg. The length of the longer leg is √3 times the length of the shorter
leg.
S
60°
S√3
2S
30⁰ So in other words as long as we know one of the sides of
these triangles we can find the other two. The 's" in both of
these pictures stands for the the short side. These two
triangles are very important to memorize. We will use them all
semester and they are frequently used in other math classes. Examples 1. Find the missing sides in each special right triangle.
We are given the hypotenuse of a 30-60-90 triangle.
The first thing we do is see what information we are
given. In this case we are given that the hypotenuse is
4. If you look at the 30-60-90 Theorem above and the
ratio picture from the notes, the hypotenuse is labeled
with a 2s. This means it is two times the short side. So
we can write a little equation to help us solve for the
short side.
60°
4
a
30°
b
4 = 2s
2=S
Now that we have the short side, we can find the other
side as well. 's' is the short side (we can visually see this
and it is always across from the 30° angle.) So s=2
If we again look at the ratio picture above, the side
opposite the 60°(the long leg), the side is labeled s√3.
So if we plug in 7 for s, we get that b=2√3. 2. Find the missing sides in each special right triangle.
45°
m
9
45°
n
We are given a 45-45-90 triangle and we know that
one of the legs is 9 units. If we look a the ratio picture
for 45-45-90 triangle in the notes, we see that the legs
are both label 's'. That means they are both the same
length. So we know that n=9. If we look back again at
the notes we see the hypotenuse is labeled s√2 and
the legs are labeled s. So since our legs are 9 units,
s=9. Now all we have to do is plug in 9 fors and we
get 9√2=m 3. Find the missing sides in each special right triangle.
Step 1:
For this example we know we have a
30-60-90 triangle and the long side
(opposite of the 60° angle) is 18.
Again look at the notes and see that
the long side of the 30-60-90 triangle
is labeled s√3. To find s we will write
a little equation:
X
60°
18
y
30°
s√3
s√√3
√3
S=
= 18
18
√3
18
√3 3. Find the missing sides in each special right triangle.
Step 2:
We have now solved for s however
we should never leave a radical in
the denominator. So let's rationalize
the denominator.
X
60°
18
y
30°
18
√3
.
√3
√3
18√/3
3
=
6√/3
So s-6√3. Remember 's' stands for
the short side. So x=6√3. 3. Find the missing sides in each special right triangle.
Step 3
Now y is the hypotenuse and we see
from the notes that the hypotenuse is
2x. So we will take s and multiple it
by 2.
X
60°
18
y
30°
6√32 = 12√3 = y

Geometry: Special Right Triangles

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Geometry: Special Right Triangles

Similar Content

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying