Special Right Triangles
30-60-90
Objective
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:
REVIEW OF 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:
The 30-60-90 Triangle
Now that we’ve reviewed over the easiest special triangle, let’s get into the harder one.
So the 30-60-90 triangle follows the same kind of pattern that the 45-45-90 triangle follows, except it takes a little more work to show.
So, let’s get into it:
Proving the 30-60-90 triangle
So, before you lose it because you saw the “P” word, don’t worry, this PowerPoint will prove it for you.
But it may take a few slides. �So, let’s get started:
First, let’s start off with an equilateral (equal sided) triangle:
Now, we can see that we have all 60 degree angles.
But, to get a 30 degree angle, we need to divide one of those 60 degree angles in half.
So, let’s drop an altitude, which will act like an angle bisector, and we’ll have two 30 degree angles instead.
By dropping the altitude, we created two right angles, AEC, and BEC.
The altitude also bisects the segment, making E the midpoint of AB and dividing AB into two equal segments, AE, and EB, both of which are half of AB.
Or in other words, AE and EB = 1.5 since AE = 3.
Now, what we wanted though, is to come up with a more generic rule, so to help us find out some rules, let’s let:
AE = t
But if, AE = t, and AC is twice as big as AE, then we know that AC = 2t
t
2t
So, now the issue is finding the remaining side to figure out how these triangles work.
Using the Pythagorean Theorem (again)
So, let’s look at what we had before:
t
2t
Using the Pythagorean theorem, we can see then that:
A = t
B = CE (segment CE since we don’t actually know what it is)
C = 2t
So:
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 30-60-90 degree triangle.
So, here are a few examples to show you.
Example 1:
Find the missing sides y and x:
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 30 degrees
(Since 180 - 90 – 60 = 30)
So, we know we have a 30, 60, 90 degree triangle.
Which means it will follow:
So, looking at our original triangle, we can see that:
Since one of the sides that is opposite of the 30 degree angle is 5/2, then that means the side that is opposite the 90 degree angle must be 5, since 5/2 * 2 = 5
Or, in other words:
Since 5/2 = t, and x = 2t, then x = 2(5/2) = 5
So, x = 5.
Which means y must be the last side to find.
Since y is
Then our last length should be:
t
2t
Example 2:
Find the missing sides y and x:
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 60 degrees
(Since 180 - 90 – 30 = 60)
So, we know we have a 30, 60, 90 degree triangle.
Which means it will follow:
So, looking at our original triangle, we can see that:
Since one of the sides that is opposite of the 60 degree angle is
But, since y is = 4, y = t, and x = 2t, then
x = 2(4)
x = 8
t
2t
Then that means to find the side that is opposite the 30 degree angle we will:
So y = 4.
Example 3:
Find the missing sides n and m:
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 60 degrees
(Since 180 - 90 – 30 = 60)
So, we know we have a 30, 60, 90 degree triangle.
Which means it will follow:
So, looking at our original triangle, we can see that:
The side that Is opposite the right angle = 14.
This means that 2t = 14
Since 2t = 14, to find t, we:
So, n = 7
t
2t
Example 4:
Find the missing sides y and x:
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 60 degrees
(Since 180 - 90 – 30 = 60)
So, we know we have a 30, 60, 90 degree triangle.
Which means it will follow:
So, looking at our original triangle, we can see that:
The side that Is opposite the right angle = 3.
This means that 2t = 3
Since 2t = 3, to find t, we:
t
2t
So, y = 3/2