Special Right Triangles
45-45-90
Objective
An explanation of roots
A root can be considered a backwards multiple. Basically, if you take a number, n, and multiply it by itself, n * n, the square will be another number, let’s call that m.
If you see a radical without the 2, it’s still the square root. Just like when we write:
x + 5 = 7, we don’t write 1*x + 5 = 7 (we just assume the 1 is there) so to do we assume the 2 is in the radical.
A FEW RULES
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Whoop whoop whoop whoop whoop whoop
You can also break Square roots
Simplified properly
A FEW THINGS TO GO OVER FIRST
So, before we go over the actual lesson, there are a few things to go over first.
Right Triangles – a right triangle is a triangle with a right angle. It’s important because a triangle can only have one right angle (since the combined angles add up to 180 degrees). All other angles are considered acute angles (or less than 90 degrees.)
Congruent Sides – Congruent sides have the same length, and are marked like so:
Congruent Angles – Congruent angles have the same angle measure, and are marked like so:
So why does this matter?
Well, a few reasons.
But enough about that, let’s get into the lesson!
So, to start off, we’re going to look at the simplest of special right triangles, the 45-45-90 degree triangle.
What’s important about 45-45-90 degree triangles is they have two congruent sides because of the two congruent angles (since each angle has an opposite side).
This allows us to discover basically everything we can about the triangle with very few parts of it given to us.
But, to find the remaining parts of the triangle, we’re going to need to use the Pythagorean theorem to set up how.
SO, WHAT IS THE PYTHAGOREAN THEOREM?
This is the Pythagorean theorem:
Where each side of a triangle is: a, b, and c; and where a and b are the sides of the triangle with c as the hypotenuse, then:
Now, we can actually see this is true when looking at an actual triangle with measurements.
Like:
Now let’s plug in these numbers and see what we get:
a = 3
b = 4
c = 5
So what does this have to do with our triangles?
Actually, by using the Pythagorean theorem, we can develop all sorts of things about these triangles.
To start off, let’s look at a 45-45-90 triangle:
Now, let’s explore this triangle a little (no matter how weird that may seem).
Now we know that the opposing sides of the 45 degree angles are congruent,
So, let’s mark that:
Now, again, since we know that those two sides are the exact same length
Then we can actually label them as well, so let’s let the length of those two
Sides be x.
x
x
Now all we are missing is the hypotenuse, so let’s use the Pythagorean Theorem to find it!
Remember, the Pythagorean theorem is:
Since we are looking for the hypotenuse, that means that a = x, and b = x, so what we have is:
And since we are trying to solve for c, let’s take the root!
And that’s essentially it!
So, now that we can see what the triangle equals, we can basically figure out anything we need when we are given a 45-45-90 degree triangle.
So, here are a few examples to show you.
Example 1:
Find the missing sides a and b:
So, first things first, we need to see what kind of triangle we have.
Well, we know it’s a right triangle because of:
And we know that the remaining angle must be 45 degrees
(Since 180 - 90 – 45 = 45)
So, we know we have a 45, 45, 90 degree triangle.
Which means it will follow:
x
x
So, looking at our original triangle, we can see that:
Since one of the sides that is opposite of a 45 degree angle is 3, then that means the other side that is opposite of the other 45 degree angle must also be 3.
So, b = 3.
Which means a must be the last side to find.
Since a is
Then our last length should be:
Example 2:
Find the missing sides x and y:
So, first things first, we need to see what kind of triangle we have.
Well, we know it’s a right triangle because of:
And we know that the remaining angle must be 45 degrees
(Since 180 - 90 – 45 = 45)
So, we know we have a 45, 45, 90 degree triangle.
Which means it will follow:
x
x
So, looking at our original triangle, we can see that:
The hypotenuse is
And since we know the hypotenuse for our bottom triangle is
Then our x = 9
Which means:
x = 9
y = 9
Example 3:
Find the missing sides x and y:
So, first things first, we need to see what kind of triangle we have.
Well, we know it’s a right triangle because of:
And we know that the remaining angle must be 45 degrees
(Since 180 - 90 – 45 = 45)
So, we know we have a 45, 45, 90 degree triangle.
Which means it will follow:
x
x
So, looking at our original triangle, we can see that:
The hypotenuse is
6
Remember, the hypotenuse from our bottom triangle is
This complicates things a little more, but we can do it none the less.
Which means:
But remember, we can’t have a radical in the
denominator, so:
Which we can simplify to:
Example 4:
Let’s try one last tough one
Find the missing sides u and v:
So, first things first, we need to see what kind of triangle we have.
Well, we know it’s a right triangle because of:
And we know that the remaining angle must be 45 degrees
(Since 180 - 90 – 45 = 45)
So, we know we have a 45, 45, 90 degree triangle.
Which means it will follow:
x
x
So, looking at our original triangle, we can see that:
The hypotenuse is
And remember, the hypotenuse from our bottom triangle is
Which means:
But remember, we can’t have a radical in the
denominator, so:
Which we can simplify to: