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Right Triangles

Okay now we can talk about right triangles. Right triangles are triangles that contain one right angle, that is, an angle of 90 degrees. Obviously, each of the other two angles must be acute, that is, less than 90 degrees. That’s the only way that they would all add up to 180 degrees.

First, we need to learn two important terms associated with right triangles. The two sides that meet at the right angle are called legs, and the side opposite the right angle is called the hypotenuse. So, we mentioned legs briefly in the previous video, but here’s the formal definition of legs. And also you need to know the word hypotenuse.

So we have two legs here in a right triangle and then the long side is the hypotenuse. In this triangle, AC and BC are the legs, and AB is the hypotenuse. The hypotenuse is always opposite the largest angle. A 90 degree angle is always going to be the largest angle. And so the hypotenuse is always the longest side of a right triangle.

The three sides of a right triangle are related by one of the most important theorems in all of mathematics: the Pythagorean theorem. This theorem is attributed to Mr. Pythagoras, who lived back in BC times. And this theorem is the special property that separates all right triangles from all non-right triangles. So here’s the amazing theorem.

If the triangle’s a right triangle then, a squared plus b squared equals c squared. Notice, this, this works only for right triangles. If the triangle is not right, if the angle is close to being right. For example, if the angle is 89.99 instead of 90, then this theorem does not work. Notice that this is leg squared plus leg squared equals hypotenuse squared. It’s lengthier to say that but notice that what we’re doing we’re squaring the two legs adding them together and that equal the hypotenuse squared.

Now that very important because we can change the letters around but it won’t change that relationship. Notice also this is a two way theorem. What do I mean by that? If we know the triangle’s a right triangle, then we can use the formula a squared plus b squared equals c squared.

If we know the formula, a squared plus b squared equals c squared, works with the sides of a triangle, then we can deduce. That the angle opposite the c is a right angle. So, we can use this. Either you can use the right angle to deduce that we can use the formula. Or if we’re, if the formula works we can deduce that we have a right angle.

It works either way. Here’s a practice problem. Pause the video. And then we’ll talk about this. Okay, those of you who are familiar with Pythagorean triplets, you may see a shortcut here.

Pretend we don’t know about the Pythagorean triplets for a second. Let’s just follow the ordinary logic of the Pythagorean theorem here. So, just for the practice, we know that a squared plus b squared equals c squared. We’ll plug in a equal 6, b equals 8. We’ll plug these in. Square them, 36 plus 64 equals 100.

So if c squared equals 100, the square root of that would be 10. And what we have is this 6, 8, 10 triangle. And of course this is not any surprise if you remember your Pythagorean triplets. We’ll talk about those in a moment, if you aren’t familiar with that term. Here’s another practice problem. Pause the video and then we’ll talk about this.

So some people might think, well gee this is the same problem over again. We got six and eight so of course c equals 10. But look closely at the triangle. Right now, b is the hypotenuse, not c. And so b is the longest side.

In fact c is the shortest side and so there’s no way c can be ten. There’s now way it can be longer than the hypotenuse. So the problem here is that here the letters have been flipped around. Now this is a little bit devious. It, it’s debatable whether the test would do this to you, but theoretically this would be fair game.

You can’t just believe in the letters. You have to understand the relationships. What the Pythagorean theorem says is that leg squared plus leg squared equals hypotenuse squared. So technically, in this triangle b squared equals a squared plus c squared. And so if we wanna solve for c squared we have to subtract a squared, c squared equals b squared minus a squared.

8 squared minus 6 squared 64 minus 36 which is 28. Take the square root of that. Of course we remember from previous modules how to simplify a square root. We can simplify that to 2 root 7, and that is the length of c. That is the simplified form that would be listed on the test. Again, if you are not familiar with the idea of simplifying square roots, I suggest go back to the power and roots module where this is discussed in depth.

Here’s another one involving roots. Pause the video and then we’ll talk about this. As might be apparent, the Pythagorean theorem really allows for no end of practice with square roots and operations with square roots.

And again, if this is stuff that is unfamiliar to you. Go back to the Power and Roots video and watch the, the mod, watch the video on Operations with Roots. That’s exactly what we’re doing here. So here we have the bonafide Pythagorean theorem, a squared plus b squared equals c squared.

We’re gonna square those two radicals. Of course square root of 7 squared is just 7. 2 root 5 squared, well, that’s 2 squared times root 5 squared. So that’s 4 times 5, which is 20 and so that adds up to 27. And so c squared. C should equal the square root of 27.

We can simplify that because, of course, that’s 9 times 3. Square root of 9 is 3. So this simplifies to 3 root 3. So this would be a triangle where all three sides are radical expressions. That happens sometimes. It can be very helpful to know the sets of integers that satisfy the Pythagorean theorem equation.

These sets of three integers are called Pythagorean triplets, and knowing them can be a huge time-saver on the test. The simplest Pythagorean triplet is 3,4,5. So here’s a 3,4,5 triangle. Notice that we don’t even need to draw the perpendicular symbol. Just by the fact that the side satisfy the equation, a squared plus b squared equals c squared.

That’s enough to guarantee. We absolutely know that we have a right angle there. Because those numbers obey the formula. Two other Pythagorean triplets that the test likes are 5, 12, 13, and 8, 15, 17. Those are good ones to memorize. Sometimes, on advanced questions, they will also use 7, 24, 25.

That one is rare, but if you really want to be safe you should memorize that one also. Notice we could also multiply any of these fundamental triplets by any number to create a new set of three numbers. So starting with 3, 4, 5, we could multiply that by 2, and we get 6, 8, 10. Multiply by 3, and get 9, 12, 15 by 4, by 5, by 6, by 7, by 8, and so forth.

Similarly we could multiply by multiples of 5, 12, 13 oe 6, 15, 17. So you really don’t have to memorize all the multiples. All you have to do is memorize those starting ones. Those four starting triplets. And then you can easily find the multiples if you need them. Now this leads us to a discussion of proportional reasoning in right triangles.

The test could give you a right triangle with two sides that are relatively large numbers. So technically if we wanted to find x we’d have to do 24 squared, plus 45 squared equals x squared. In this problem it would be a huge mistake to square 45. That would be some really big number, square 24, that would be another big number.

Add those two numbers together and try to find the square root of the resulting four digit number. That would be a spectacularly bad idea. Instead find the greatest common factor of the sides and factor that out. Well the greatest common factor of 24 and 45 is three. Okay well think about it this way suppose we scale down that triangle.

By a factor of three, a scale factor of three. So, we have the large triangle, our starting triangle, and then we have a scaled down version with the unknown hypotenuse y. Well, of course that scaled down version, that looks pretty good, because that’s actually one of our Pythagorean triplets. Of course that’s the 8, 15, 17 triplet we don’t even have to do any calculations.

We can see instantly well of course why this has to equal 17. Well once we know know that then it’s gonna be very easy to find x we just have to scale back up. We got to smaller one by dividing by 3. We get to the larger one by multiplying by 3. So x should just equal 3 times 17, which is 51.

And so, basically, with almost no calculations. Really, the only calculation we did here is 3 times 17. That was enough to solve this problem. We never had to deal with four digit numbers. That’s really important. Basically, almost any time on the test that you’re finding yourself dealing with four digit numbers that are not numbers given in the problem.

Chances are you’re making your life much harder than it has to be. It may be that when we factor out the greatest common factor we will get one of our time-saving Pythagorean triplets. Otherwise, it may be that we just wind up with a small triangle with some very easy numbers, and we can just quickly apply the Pythagorean triplet.

In that much, in the Pythagorean theorem, in that much smaller triangle. So here’s a problem. Pause the video and we’ll talk about this. Well the greatest common factor of 36 and 72 is 36. If we divide all the sides by 36 we get a triangle with legs of 1 and 2.

Well those are very simple numbers. If we have a equals 1 and b equals 2, plug this into the Pythagorean theorem we get c squared equals 5, c equals square root of 5. And so that that means in the big triangle all we have to do is multiply square root of 5 times 36. Multiply it back by that greatest common factor.

And so RS equals 36 root 5. So notice we did our Pythagorean theorem with the small numbers. Very convenient. We never had to square 36 and square 72. That would just be a disaster. So you don’t wanna square gigantic numbers.

You wanna divide them down by a greatest common factor, and then work with much smaller numbers. Here’s another practice problem. Pause the video, and then we’ll talk about this. Okay, so obviously from what we’ve been talking about, we do not want to get into squaring 42 and squaring 56, and getting those huge numbers.

We don’t wanna do that. Instead, we find the greatest common factor which is 14, and 56 is 4 times 14, 42 is 3 times 14. So we’ll just scale down to we have a leg of three and a hypotenuse of four. Notice this is not a 3,4,5 triangle cause the hypotenuse is 4. Instead what we have is b squared equals 4 squared minus 3 squared.

16 minus 9 wich is seven. So b squared would be the square root of 7. And then we just, just scale back up by the factor of 14. The greatest common factor is the scale factor. We scale back up. And we get that x equals 14 root 7.

In summary, right triangles have one 90 degree angle and two acute angles. Right triangles always have one hypotenuse, the longest side, and the two legs. The legs are the sides that touch the right angle. The Pythagorean Theorem tells us that a squared plus b squared equals c squared. That’s only true for right triangles, and in fact we know if it’s a right triangle we can use that formula.

And if the formula is true we can deduce that we have a right angle. Works both ways. We’ll be making extensive use of this theorem in the remaining geometry videos. The Pythagorean triplets, these are very handy to have memorized because they save you a lot of time. If you know these, you can avoid all kinds of unnecessary calculations.

And finally, if the sides given are larger, the sides of the triangle given are relatively large numbers, divide down by the greatest common factor. Do the computations in the smaller triangle and then scale back up. That is a much easier way to think about these problems.

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