Geometry: Right Triangles

In the past two videos we discussed basic elements of triangles, then the differentiation between different types of triangles. Now we are going to focus on right triangles.

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Transcript: 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 a 180 degrees.

Right Triangles: Legs and Hypotenuse

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 are two legs here in a right triangle and then the long side is the hypotenuse.

right triangles, hypotenuse - magoosh

Image by Pyty

This triangle, AC and BC are the legs, and AB is the hypotenuse. The hypotenuse is always opposite the largest angle. The 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 Pythagorean Theorem

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 triangles are right triangle, then a squared + b squared = c squared. Notice that this works only for right triangles.

If the triangle’s not right, if the angle is close to being right, for example, 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 are doing is we are squaring the two legs, adding them together, and that equals the hypotenuse-squared. Now that is very important because of course we can change the letters around, but it won’t change that relationship.

Two-Way Theorem

Notice also this is a two-way theorem, what do I mean by that? If we know the triangle is a right triangle, then we can use the formula a squared + b squared = c squared. If we know the formula a squared + b squared = c squared works for 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 the formula works we can deduce that we have a right angle. It works either way.

Right Triangles: Practice Problem One

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 thee practice, we know that a squared + b squared = c squared, we’ll plug in a = 6, b = 8.

We’ll plug these in, square them, 36+ 64 = 100. So c squared = 100, the square root of that would be 10. And what we have a is 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.

Right Triangles: Practice Problem 2

Here’s another practice problem, pause the video and then we’ll talk about this.

right triangles, pause - magoosh

Image by WhiteBarbie

So some people might think well gee, this is the same problem over again. We got 6 and 8 so of course c = 10, but look closely at the triangle. Right now b is the hypotenuse not c, and so a is the longest side. In fact, c is the shortest side. And so there’s no way that c could be 10.

There’s no way it could 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’s debatable whether the test would actually 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.

And what the Pythagorean theorem says is that leg squared plus leg squared equals hypotenuse squared. So technically, in this triangle b squared = a squared + c squared. And so if we wanna solve for c squared, we have to subtract a squared, c squared = b squared- a squared, 8 squared- 6 squared, 64- 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’re 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.

Right Triangles: Practice Problem Three

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 roots video and 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 + b squared = c squared.

We’re going to square this to radicals. Of course square root of 7 squared is just 7. 2 root 5 squared, well that’s 2 squared times root 5 squares so that’s 4 times 5 which is 20 so that adds up to 27. So c squared, c should equal the square root of 27, we should be able to 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 that 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 + b squared = 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 wanna 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 get 6, 8, 10 multiply it 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 or 8,15,17. So you really don’t have to memorize. For 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 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 have to do 24 squared + 45 squared = 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 resultant 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 3. 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 triangle, we don’t even have to do any calculation. We can see instantly, well of course y just has to equal 17.

Once we know that, then this can be very easy to find x. We just have to scale back up. So we got to the smaller one by dividing by 3, we get to the larger one by multiplying by 3. So x should just equal 3*17, which is 51. And so basically with almost no calculations, really the only calculation we did here is 3*17. That was enough to solve this problem.

We never had to deal with four digits numbers, that’s really important. Basically, almost any time on the test that you’re finding yourself dealing with four-digit numbers that are 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 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 the sides by 36, we get a triangle with legs of 1 and 2 well, those are very simple. If we add a = 1 and b = 2, plug this into the Pythagorean Theorem, we get c squared = 5, c = square root of 5. And so 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 = 36 root 5.

So notice we did our Pythagorean theorem with the small numbers, very convenient. We never had to square 36 of square 72, that would just be a disaster–so you don’t want to square gigantic numbers. You wanna divide them down by the greatest common factor and then work with much smaller numbers.

Right Triangles: Practice Problem Four

Here is 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 wanna 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*14, 42 is 3*14.

So we’ll just scale down to, we have a leg of 3 and a hypotenuse of 4, notice this is not a 3, 4, 5 triangle, because the hypotenuse is four. Instead, what we have Is b squared equals 4 squared- 3 square, 16- 9 which is 7. So b squared would be the square root of 7. And then we just scale by up by the factor of 14, the greatest common factor is the scale factor.

We scale back up and we get that x = 14 root 7.

Summary

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 +b squared = 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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