Mathematics II (2018 edition)
- Intro to the Pythagorean theorem
- Intro to the Pythagorean theorem 2
- Pythagorean theorem word problem: fishing boat
- Pythagorean theorem example
- Pythagorean theorem word problem: carpet
- Use Pythagorean theorem to find right triangle side lengths
- Pythagorean theorem challenge
- Introduction to the Pythagorean theorem
- Pythagorean theorem II
More Pythagorean Theorem examples. Introduction to 45-45-90 triangles. Created by Sal Khan.
Want to join the conversation?
- At0:46and on all the exercise problems, why do they claim the shorter leg to be longer than the oter one when it is actually smaller? That really confused me.(21 votes)
- See, it does not matter which leg is smaller or bigger.
Only the hypotenuse matters -
Example - A triangle with sides of measurement 3,4 & 5 than you can know that the hypotenuse is the longest leg.
But if we had to find hypotenuse than we can add in any way but in subtraction we have to subtract the smaller length from bigger!(18 votes)
- how do you get a 5 square root of 2??(22 votes)
- 50's factors are: 2*25, but 25 factors out more so 2*5*5, then as 2 of the 5's in the square root sign are same, then take it out of the square root and you are left with 5 root 2~ :)(20 votes)
- I don't understand the part when he said "The part that they don't share are equivalent" I need help on this.(10 votes)
- Each angle touches two sides of the triangle. If two angles are equal, the two sides other than the side that both those angles touch, are equal. If you still don't understand, you should watch the video again.(9 votes)
- I hope this is the right place to post this question:
How do I find the legs of a triangle using only the hypotenuse and the area?(2 votes)
- This is a fairly easy, but very tedious problem. Here is how to do it:
Use simultaneous solution from the area formula and the Pythagorean theorem
a² + b² = c²
½ ab = A (area)
ab = 2A
b = 2A/a
b² = [2A/a]²
b² = 4A²/a²
Substituting into Pythagorean Theorem:
a² + b² = c²
a² +4A²/a² = c²
a⁴ + 4A² = c²a²
a⁴ - c²a² + 4A² = 0
Let x = a²
x² - c²x + 4A² = 0
x =½ [c² ±√(c⁴-16A²)]
a² = ½ [c² ±√(c⁴-16A²)]
This will give you two answers. One of these will, of course, be a². The other will be b². Just take the positive square roots of those two answers to get the two legs' lengths.
Example. A right triangle has a hypotenuse of 5 cm and an area of 6 cm². What are the lengths of the legs?
a² = ½ [c² ±√(c⁴-16A²)]
a² = ½ [5² ±√(5⁴-16(6)²)]
a² = ½ [25 ±√(625-16(36))]
a² = ½ [25 ±√(625-576)]
a² = ½ [25 ±√(49)]
a² = ½ [25 ± 7]
a² = ½  OR a² = ½ (32)
a² = 9 OR a² = 16
a = 3 OR a = 4
Thus the two leg's lengths are 3 cm and 4 cm.(17 votes)
- When Sal said that the lengths of a and b are both equal to 5 because the angles of 45 degrees in the triangle are the same, isn't that true for all right triangles? Because right triangles always have 90degree angle with 45degree angle and another 45degree angle right? And if that was true, why aren't all of the right triangles sides a and b the same?(1 vote)
- No, because if you have a longer and shorter leg, you can have a 30-60-90 triangle, which is still a right triangle.(4 votes)
- How can the square root of 50 turned to the squared root of 2 times 5(2 votes)
- A 90 degree angle always has two 45 degree angles and a 90 degree angle. You said that if 2 angles are equal in a triangles then the sides they are not sharing with each other are equal as well. But in my math textbook it says that sides of right angled triangle are always in ratio 3:4:5.(1 vote)
- The sides of a right triangle are not always in the ratio 3 : 4 : 5. That is just one example of a primitive Pythagorean triple. There are an infinite number of possible ratios for right triangles and 3 : 4 : 5 is just one possibility.(3 votes)
- Hi how do you find c squared is given then how do you find b squared(1 vote)
- How does sine cosine and tangent really have anything to do with right triangles?(1 vote)
- That's a really good and important question!! Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.
for more info--
I promised you that I'd give you some more Pythagorean theorem problems, so I will now give you more Pythagorean theorem problems. And once again, this is all about practice. Let's say I had a triangle-- that's an ugly looking right triangle, let me draw another one --and if I were to tell you that that side is 7, the side is 6, and I want to figure out this side. Well, we learned in the last presentation: which of these sides is the hypotenuse? Well, here's the right angle, so the side opposite the right angle is the hypotenuse. So what we want to do is actually figure out the hypotenuse. So we know that 6 squared plus 7 squared is equal to the hypotenuse squared. And in the Pythagorean theorem they use C to represent the hypotenuse, so we'll use C here as well. And 36 plus 49 is equal to C squared. 85 is equal to C squared. Or C is equal to the square root of 85. And this is the part that most people have trouble with, is actually simplifying the radical. So the square root of 85: can I factor 85 so it's a product of a perfect square and another number? 85 isn't divisible by 4. So it won't be divisible by 16 or any of the multiples of 4. 5 goes into 85 how many times? No, that's not perfect square, either. I don't think 85 can be factored further as a product of a perfect square and another number. So you might correct me; I might be wrong. This might be good exercise for you to do later, but as far as I can tell we have gotten our answer. The answer here is the square root of 85. And if we actually wanted to estimate what that is, let's think about it: the square root of 81 is 9, and the square root of 100 is 10 , so it's some place in between 9 and 10, and it's probably a little bit closer to 9. So it's 9 point something, something, something. And that's a good reality check; that makes sense. If this side is 6, this side is 7, 9 point something, something, something makes sense for that length. Let me give you another problem. [DRAWING] Let's say that this is 10 . This is 3. What is this side? First, let's identify our hypotenuse. We have our right angle here, so the side opposite the right angle is the hypotenuse and it's also the longest side. So it's 10. So 10 squared is equal to the sum of the squares of the other two sides. This is equal to 3 squared-- let's call this A. Pick it arbitrarily. --plus A squared. Well, this is 100, is equal to 9 plus A squared, or A squared is equal to 100 minus 9. A squared is equal to 91. I don't think that can be simplified further, either. 3 doesn't go into it. I wonder, is 91 a prime number? I'm not sure. As far as I know, we're done with this problem. Let me give you another problem, And actually, this time I'm going to include one extra step just to confuse you because I think you're getting this a little bit too easily. Let's say I have a triangle. And once again, we're dealing all with right triangles now. And never are you going to attempt to use the Pythagorean theorem unless you know for a fact that's all right triangle. But this example, we know that this is right triangle. If I would tell you the length of this side is 5, and if our tell you that this angle is 45 degrees, can we figure out the other two sides of this triangle? Well, we can't use the Pythagorean theorem directly because the Pythagorean theorem tells us that if have a right triangle and we know two of the sides that we can figure out the third side. Here we have a right triangle and we only know one of the sides. So we can't figure out the other two just yet. But maybe we can use this extra information right here, this 45 degrees, to figure out another side, and then we'd be able use the Pythagorean theorem. Well, we know that the angles in a triangle add up to 180 degrees. Well, hopefully you know the angles in a triangle add up to 180 degrees. If you don't it's my fault because I haven't taught you that already. So let's figure out what the angles of this triangle add up to. Well, I mean we know they add up to 180, but using that information, we could figure out what this angle is. Because we know that this angle is 90, this angle is 45. So we say 45-- lets call this angle x; I'm trying to make it messy --45 plus 90-- this [INAUDIBLE] is a 90 degree angle --plus is equal to 180 degrees. And that's because the angles in a triangle always add up to 180 degrees. So if we just solve for x, we get 135 plus x is equal to 180. Subtract 135 from both sides. We get x is equal to 45 degrees. Interesting. x is also 45 degrees. So we have a 90 degree angle and two 45 degree angles. Now I'm going to give you another theorem that's not named after the head of a religion or the founder of religion. I actually don't think this theorem doesn't have a name at. All It's the fact that if I have another triangle --I'm going to draw another triangle out here --where two of the base angles are the same-- and when I say base angle, I just mean if these two angles are the same, let's call it a. They're both a --then the sides that they don't share-- these angles share this side, right? --but if we look at the sides that they don't share, we know that these sides are equal. I forgot what we call this in geometry class. Maybe I'll look it up in another presentation; I'll let you know. But I got this far without knowing what the name of the theorem is. And it makes sense; you don't even need me to tell you that. If I were to change one of these angles, the length would also change. Or another way to think about it, the only way-- no, I don't confuse you too much. But you can visually see that if these two sides are the same, then these two angles are going to be the same. If you changed one of these sides' lengths, then the angles will also change, or the angles will not be equal anymore. But I'll leave that for you to think about. But just take my word for it right now that if two angles in a triangle are equivalent, then the sides that they don't share are also equal in length. Make sure you remember: not the side that they share-- because that can't be equal to anything --it's the side that they don't share are equal in length. So here we have an example where we have to equal angles. They're both 45 degrees. So that means that the sides that they don't share-- this is the side they share, right? Both angle share this side --so that means that the side that they don't share are equal. So this side is equal to this side. And I think you might be experiencing an ah-hah moment that right now. Well this side is equal to this side-- I gave you at the beginning of this problem that this side is equal to 5 --so then we know that this side is equal to 5. And now we can do the Pythagorean theorem. We know this is the hypotenuse, right? So we can say 5 squared plus 5 squared is equal to-- let's say C squared, where C is the length of the hypotenuse --5 squared plus 5 squared-- that's just 50 --is equal to C squared. And then we get C is equal to the square root of 50. And 50 is 2 times 25, so C is equal to 5 square roots of 2. Interesting. So I think I might have given you a lot of information there. If you get confused, maybe you want to re-watch this video. But on the next video I'm actually going to give you more information about this type of triangle, which is actually a very common type of triangle you'll see in geometry and trigonometry 45, 45, 90 triangle. And it makes sense why it's called that because it has 45 degrees, 45 degrees, and a 90 degree angle. And I'll actually show you a quick way of using that information that it is a 45, 45, 90 degree triangle to figure out the size if you're given even one of the sides. I hope I haven't confused you too much, and I look forward to seeing you in the next presentation. See you later.