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The Pythagorean theorem intro | Right triangles and trigonometry | Geometry | Khan Academy
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50 Pythagorean theorem Worksheet with Answers
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What is the Pythagorean Theorem?
Hello and welcome to this review of the Pythagorean Theorem! We’ll cover how to use it properly, and also cover some special triangles called “Pythagorean triples” that might save you some time when running tests. So let’s get started!
Formula of the Pythagorean theorem
First things first: what is the Pythagorean theorem? The Pythagorean theorem is \(a^2+b^2=c^2\).
Well this is used to find the length of one side of a right triangle when we know the lengths of the other two sides. The triangle must be a right triangle, meaning it has an angle that measures exactly 90 degrees, like this one:
The theorem is very easy to remember and just as easy to apply! In the theorem, \(a\), \(b\), and \(c\) are the lengths of the three sides of the triangle. But what is what? Let’s start by figuring out where to find \(a\), \(b\), and \(c\) in a triangle.
First you can tell that you are dealing with a right triangle because you can see this little square in one of the angles. This is the symbol for a right angle (or 90˚ angle). Any triangle that has a right angle must be a right triangle.
So now we need to decide where to place the three side lengths. The key is to start with \(c\) which is always on the side opposite the right angle.
This is called the hypotenuse and is always the longest side.
You may be wondering, “If \(c\) is always opposite the right angle, how can I tell which of the other two is \(a\) and which is \(b\)?” It’s a good question, and the answer is, it doesn’t matter! Either of these two sides, called legs, can be used as \(a\) and then just the other for \(b\). Let’s take 3 cm for \(a\) and use 4 cm for \(b\). Our sentence looks like this when we have filled it in: \(3^2+4^2=c^2\)
Now we can only evaluate 3 squared and 4 squared, which means we multiply 3 times 3 and 4 times 4 to get:
\(3 \times 3=9\hspace{30px}4\times 4=16\)
\(9+16=c^2\)
So what is the final answer for \(c^2\)? A little addition tells us.
\(9+16=25\)
\(25=c^2\) or \(c^2=25\)
This is where it gets a little tricky. We now know that \(c^2=25\), but we want to know what \(c\) is, not \(c\) squared! So how can we get rid of this little 2? Well we use the inverse or opposite operation of squaring something! And this inverse operation is the square root! Since this is an equation, whatever we do to one side of the equation we must do to the other, so I’m going to take the square root of both sides.
\(\sqrt{c^2}=\sqrt{25}\)
The square root of \(c^2\) is \(c\) and the square root of 25 happens to be 5. Look at this. It’s always good to check our answer to see if it makes sense.
Since we’re finding \(c\), it should be longer than either of the other two sides, and 5 is greater than 4 or 3. Also because of the triangle inequality theorem, which we’ll get to in a later video, the hypotenuse has to be smaller than the sum of the other two sides, which means 5 must be less than 3+4. What is it.
So you may have noticed that the answer to this problem was a nice, neat integer (5). This is actually pretty rare if we look at random triangles. But it’s not uncommon for a math problem you might see on a test. It happens whenever a problem uses a Pythagorean triple. The triangle we just looked at is the most common type, a 3-4-5 right triangle. The legs measure 3 and 4 and the hypotenuse is 5.
Sometimes it is disguised by multiplying all numbers by 2, meaning we get a (6-8-10), or multiplying by 10, meaning we get a (30-40-50) length or another have number. You don’t need to know this to solve a Pythagorean theorem problem, but it’s a nice shortcut to save you some time or allow you to check your answer in another way. Other Pythagorean triples are 5-12-13, 8-15-17 and 7-24-25. There are many more, but these are the ones you’ll see the most.
Of course, there are right triangles that are not Pythagorean triples. Let’s look at one so we can see what the answer will look like.
Again we solve for the hypotenuse (the longest side opposite the 90˚ or right angle). This is the length \(c\). Substituting 5 and 7 for \(a\) and \(b\), we get the following:
\(5^2+7^2=c^2\)
\(2^5+4^9=c^2\)
So let’s add 25 + 49 to get \(74=c^2\).
Then we take the square root of each side and find that \(c = \sqrt{74}\) even though 74 is not a perfect square now. That is the answer. Taking the square root on a calculator only gives you an approximation of the result (which is roughly 8.60232526…). We can use this approximation to do our checks: it’s larger than each of the other two sides and smaller than the two sides combined. But when writing the answer, we should use the square root form.
Thanks for watching and happy learning!
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