The Pythagorean Theorem

In this lesson we go over how to use the Pythagorean theorem to solve for missing side lengths in right triangles. The slides go over how Pythagoras discovered the pattern for his formula as well as how to use it to find a missing hypotenuse or modify the Pythagorean theorem to find a missing leg. Then we move on to utilizing the converse and Pythagorean triples to determine if a triangle is right and how to quickly find missing pieces in our head without all the squaring!

So without further ado, read through the slides below to get a feel for how to use the Pythagorean theorem!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

The Coordinate Plane

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • The Pythagorean Theorem
  • And its converse
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Utilize the Pythagorean theorem to solve problems
  • Utilize the inverse of the Pythagorean theorem to identify right triangles
  • Identify and utilize Pythagorean triples
  • Right Triangles
  • From the name of this slide you’ve probably guessed we will be working with right triangles. So let’s see one:
  • What is it that makes this triangle special?
  • This box indicates that this is a right angle, or a 90 degree angle.
  • Because it has a right angle, this particular triangle is classified as a right triangle.
  • This shape occurs everywhere from something leaning against a wall, to engineering structures, and even to distances between points!
  • Right Triangles
  • Now we can dive into the different parts of a right triangle.
  • First, we always have a right angle, which we pointed out in the last slide:
  • Next, we have two legs which are the sides adjacent to, or touching, the right angle:
  • Then finally, the long side opposite the right angle is called the hypotenuse:
  • The hypotenuse will always be the longest side of a right triangle!
  • The Pythagorean Theorem
  • So what is all this Pythagorean stuff?
  • It is a special relationship between all the sides.
  • Formally, the Pythagorean theorem is stated as “the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse.”
  • Doesn’t that sound lovely.
  • Perhaps you’ve heard it stated something more like this:
  • More familiar but still not clear? How about a visual:
  • Proof
  • So how did Pythagoras discover this unique relationship?
  • Well, let’s dissect his theorem.
  • We are squaring our side lengths, so why not draw some squares on the sides:
  • The length of our left side is 3:
  • And we said we drew a square so both its sides are the same:
  • Then the other two sides would make squares that followed the same pattern:
  • Proof
  • So what is the area of each of these leg squares?
  • Remember, we said earlier that we have to take the sum of the legs, so what is 9+16?
  • 25, which is the area of the square attached to our hypotenuse!
  • This is the pattern Pythagoras noticed to develop his theorem for the relationship between the sides.
  • Now we can all utilize the Pythagorean Theorem when we need to find a missing side length in a right triangle!
  • An example
  • So let’s get our right triangle back:
  • Now let’s get some measurements of those legs:
  • So the problem is that we’d like to know the length of our hypotenuse.
  • What is the relationship we know between the sides?
  • Great, so what does this mean?
  • The Pythagorean theorem is just a formula!
  • The sides are the variables we plug in:
  • An Example
  • All we have to do is replace our letters with the corresponding numbers.
  • So let’s do just that!
  • 4 substitutes in for a:
  • 3 substitutes in for b:
  • Now we can solve this for our missing side!
  • An Example
  • First our exponents:
  • Then c squared comes along for the ride.
  • What is 16+9?
  • Now we have an equation to solve with a radical!
  • What is the square root of 25?
  • Thus the missing side on our triangle, the length of the hypotenuse, is 5!
  • Missing Leg
  • The hypotenuse might not be the side length we are missing, however:
  • In this example we are missing the bottom side, or b.
  • So if we plug in our numbers, it looks like this:
  • Then square what we’ve got:
  • So then, how can we figure out what b equals?
  • Back to solving equations!
  • Missing Leg
  • We want b to be by itself, so 25 is in the way.
  • What’s the opposite of adding 25?
  • Subtracting!
  • So then now we are yet again back to exponents:
  • Now all that’s left is:
  • Thus, our missing leg is 12!
  • TIP: If you don’t enjoy solving every time, instead you can remember to subtract the square of the leg from the square of the hypotenuse instead!
  • Backwards?
  • Something funny about theorems is that they prove things in two directions sometimes.
  • Hmm…What’s that mean?
  • Well, basically, the Pythagorean theorem says if you have a right triangle, then you can use the formula for the side lengths.
  • Is it true backwards?
  • If we have sides that satisfy the Pythagorean formula, does that mean we have a right triangle?
  • Yes!
  • Backwards
  • Fair enough, but why would this matter?
  • I’m so glad you asked.
  • Looks familiar, right?
  • But something is missing…
  • We don’t have our box showing this is a right triangle.
  • This is where the converse, or backwards form, of the Pythagorean theorem becomes useful.
  • We can plug our sides into the formula to see if it works (which we already know to be true from a previous example).
  • Pythagorean Triples
  • So we know that even though there was no box showing that we had a right triangle, the converse showed us that since the sides satisfy the theorem, we must have had a right angle!
  • But as always, wouldn’t it be nice to find things quicker.
  • Luckily, as lazy mathematicians, we can find a way!
  • Did you notice the two triangles we used as examples?
  • The first has sides of 3, 4, and 5.
  • The second has sides of 5, 12, and 13.
  • These are called Pythagorean triples!
  • Pythagorean Triples
  • Any set of sides that satisfies the Pythagorean theorem, works in the formula, is called a Pythagorean triple.
  • The two most common triples that students see are:
  • 3, 4, 5
  • 5, 12, 13
  • In fact, these are the two we’ve been using throughout this lesson!
  • But why does this matter?
  • To make things easier of course!
  • Pythagorean Triples
  • Take this example:
  • Rather than going through all the effort we did the first time, now we can recognize this as a 3, 4, 5 triangle.
  • The missing side, the hypotenuse, is the 5 part!
  • We know this because if 3 and 4 are two of the sides, the Pythagorean theorem will always spit out 5 for the last side, just like in our example where we solved for the hypotenuse.
  • What about this, though?
  • Well that’s not a common triple…or is it?
  • Pythagorean Triples
  • This triangle is actually similar to our 3, 4, 5.
  • That’s just a fancy way of saying that we can multiply all the sides by something to make it look like the second one!
  • Let’s take two of our corresponding sides, 4 and 8.
  • What do you notice about these numbers?
  • That’s right! 8 is just the double of 4.
  • What about 3 and 6?
  • If the pattern is true for two sides, it has to be true for the third.
  • Thus, 5×2=10, so our missing side in our second triangle is 10!
  • Pythagorean Triples
  • One last example so you can try:
  • Hmm, 36 is a big number.
  • Since 36 is a leg length, let’s try our 3, 4, 5 first:
  • 3x12=36, but 4x12 is not equal to 15, so 12 cannot be correct.
  • 4x9=36, but 3x9 is not equal to 15, so 9 cannot be correct either.
  • If our 3, 4, 5 isn’t working, how about this?
  • 12x3=36, and 5x3=15……So what is 13×3?
  • So our missing side is 39! Much easier than squaring 36 and 15!
  • Conclusion
  • A right triangle is a triangle that includes a right angle (90 degrees).
  • The two sides touching the right angle are called legs.
  • The opposite side (the longest) is called the hypotenuse.
  • The Pythagorean theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse.
  • We can use the Pythagorean theorem to find missing sides of a right triangle by plugging the sides into the formula: a^2+b^2=c^2 where a and b are the lengths of the legs and c is the length of the hypotenuse
  • To find a missing leg, use c^2-a^2=b^2.
  • Three numbers that satisfy the Pythagorean Theorem are called Pythagorean Triples
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