Never argue with a 90° triangle…It’s always RIGHT!
What is Pythagorean Theorem anyway and when do we use it in REAL life? Pythagorean Theorem is EVERYWHERE because RIGHT TRIANGLES are everywhere! This theorem is thought to have been developed by Pythagoras of Samos – however, this is somewhat controversial. Pythagoras never kept a written account of…anything! What we know about him is due in large part to other Pythagoras scholars’ writing!
At any rate, this theorem helps us to determine the lengths of the sides of right triangles. It’s a pretty genius “trick” if you ask me!
Identifying sides a, b & c!
Pythagorean theorem is basically just a big fancy way to determine the length of each side of a right triangle. A lot of times when we’re dealing with Pythagorean theorem, we’re given a right triangle and this formula:
and it’s kind of confusing at first as to what all of that means. So I’d like to show you where all of these different variables go in relation to the right triangle and then we’re going to start solving some right triangles.
So we have this right triangle:
and we can tell that it’s a right triangle because there are two sides that create a 90-degree angle. These are going to be your a and b sides. Now, it does not matter which of these sides is labeled a or b because of the commutative property – we can switch the order of numbers or variables that are being added!! We know that these two are the a and b sides because that’s where that 90-degree angle indicator is touching the two sides. These two sides make up the 90 degree angle.
Now let’s talk about the third side of the triangle. You may have heard the word hypotenuse – that side will always be labeled c. It will always be the hypotenuse. This is also the longest side of a right triangle.
So, to recap: c will always be the longest side of the right triangle, and a and b will be the two shorter sides that make up the right angle.
Let’s practice squaring each side!
Finding the square of each side is simple! Just multiply each number by itself! Let’s use this triangle as an example:
4² = 4 x 4 = 16 3² = 3 x 3 = 9 5² = 5 x 5 = 25
Be careful not to simply multiply each number by 2! It’s not the same thing!
Now let’s check to see if this *is* a right triangle or not!
Here we see a triangle with sides 6, 7 and 9 units. First, let’s label. Remember, that our LONGEST side will have to be c, and the two shorter sides are a and b.
Now, let’s substitute everything in to the Pythagorean Theorem and square each side
a² + b² = c²
6² + 7² = 9²
36 + 49 = 81
Is this true?!?! No?!?! Then it’s not a right triangle!
Now let’s try finding the missing side length of a triangle.
Here we have a triangle with sides 12 and 5 labeled, but we are missing the length of the third side. We also can see that the 12 and 5 unit sides create the right angle – that must mean that those are our two shorter sides a and b!
So let’s substitute everything in to the Pythagorean Theorem and see what we come up with!
a² + b² = c²
12² + 5² = c²
You’ll notice that I didn’t substitute anything in for the “c” side because we don’t know what it is yet. That’s what we’re going to use ALGEBRA for to find out!
12² + 5² = c²
144 + 25 = c²
169 = c²
At this point, a lot of times students will stop and think “That’s it! C = 169!” BUT c squared equals 169, not just c by itself. So now, we must find the square root!
√169 = √c²
13 = c
Now we know that the missing side length is 13 units!
We can also use this method to find missing a or b side lengths.
The steps are slightly different, but with our handy friend ALGEBRA we can figure it out!
So here we have a triangle and we are given side lengths 15 and 17. We also see that one of the side lengths is missing. AND that missing side length and the 15 side are what create our right angle, so we know that those are our a and b sides so c must equal 17!
Now, let’s substitute in everything into the Pythagorean Theorem. And be careful! You must set it up so that a² and b² are being added, not a² and c²!
15² + b² = 17²
225 + b² = 289
At this point, we need to subtract 225 from both sides of the equation.
225 + b² = 289
b² = 64
And just like before, we need to find the square root of each side
√b² = √64
b = 8
Why does this work?
Magic! No. Let’s look at a diagram.
We can see that our right triangle has sides of 3, 4 and 5. When we line up a perfect square with each side of the triangle, we can see that:
- the squared side of 3 has 9 grid-squares
- the squared side of 4 has 16 grid-squares inside of it
- the squared side of 5 has 25 grid-squares inside of it.
If we take those two smaller squares, the 16 and the 9, and we add them together, that equals 25, which is the same as the larger square!! This will work this way for every single right triangle. Try it with a piece of graph paper! (Note: You will need to cut out a square for the hypotenuse because that side of the triangle will be diagonal to your grid squares on the graph paper!) If you create a triangle with a gridded square lined up on each side of the triangle you will find that the two smaller sides will always add up to equal the one larger side!!
If you like this activity, please check out my Intro to Pythagorean Theorem Boom Cards! This contains guided step-by-step videos of each part of the process. It’s perfect for 8th-grade students who are just starting to learn about using the Pythagorean Theorem!