45-45-90 Triangles - Concept
A 45 45 90 triangle is a special type of isosceles right triangle where the two legs are congruent to one another and the non-right angles are both equal to 45 degrees. Many times, we can use the Pythagorean theorem to find the missing legs or hypotenuse of 45 45 90 triangles. The ratio of the sides to the hypotenuse is always 1:1:square root of two.
Something special in geometry
is the 45, 45, 90 triangle.
Well, a 45, 45, 90 triangle is an isosceles
right triangle where these two legs
are congruent to each other.
The reason why it's 45, 45, 90 is because
if we know that these two angles are
congruent to each other, because the
isosceles triangle theorem, then
we can say that 180 degrees is equal
to 90, plus X plus X. So if
I add these up, I'm going to have 180
is equal to 90, plus 2 X, so I'm
going to subtract 90 from both sides
and I get 90 is equal to 2X, and
then I'm going to divide by 2 to
solve for X. And 90 divided by
2 is 45, which means each of these angles
that are congruent to each other
have to be 45 degrees.
So in an isosceles right triangle you're
going to have a 45 degree, a 45 degree
and a 90 degree.
So that's we mean when
we say 45, 45, 90.
Now something is going on with
these angles and sides.
And if I wrote in that these were both X and
I would say that this is my hypotenuse
C, let's apply the Pythagorean
theorem and see what happens.
Pythagorean theorem says A squared plus
B squared equals C squared and A and
B here are both X. So I'm going to
write that X squared plus X squared
is equal to C squared.
I can combine like terms here and X squared
plus X squared is 2X squared.
So if I want to solve for my hypotenuse
C, I'm going to take the square root
of both sides, and the square root
of X squared is X, and there is no
whole number square root of 2. So
C is equal to X times the square
root of 2. Well, that's a little
difficult to understand.
So let's say we had an isosceles right triangle
with sides of length 1 and I'm
trying to find the hypotenuse.
So maybe this will make sense
with this triangle.
Here we'll have 1 squared plus 1
squared is equal to C squared
Well, 1 plus 1 is 2. So if I take the
square root of both sides, I find
that my hypotenuse is equal to the
square root of 2. So now what
I see it's talking about is if you know
the side of one of your legs, if
you know that length, you're going to.
multiply it by the square root of 2.
So to get from the leg in a 45, 45, 90.
triangle, you're going to multiply by
the square root of 2.
Let's say, however, you don't know what that
leg is. And you know the hypotenuse.
So I'm going to draw another
triangle over here.
45, 45, 90, and let's say you said this
was 3. To go from your hypotenuse
to your leg, you're going to undo multiplying
by the square root of 2.
So you're going to divide by the square
root of 2. So this answer
right here will be 3 divided by the
square root of 2.
And we can't
have a square root in our denominator here.
So now this is becoming quite a chore.
We're going to multiply by square root
of 2. Multiply by the square root
of 2. So we'll have in our numerator
3 times the square root of
2. Square root of 2 times square root of 2
is 2 because you'll have the square root
of 4. So this is actually 3 times.
the square root of 2 divided by 2.
So if we go back to our original
drawing here where we said.
for any right triangle where you have
two legs that are congruent, to go
from your leg to your hypotenuse, all
you need to do is take that number
and multiply it by the square root of
2. So if X is 5, your hypotenuse
is 5 times the square
root of 2. To go from your hypotenuse back to one of
your legs, you're going to divide by
the square root of 2.
So keep that in mind and solving for missing
sides, an isosceles right triangle
is pretty simple.