Geometry Pythagorean Theorem

Welcome back to Educator.com.0000

This next lesson, we are going to go over the Pythagorean theorem.0002

The Pythagorean theorem says that, in a right triangle, the sum of the squares of the measures of the legs0007

is equal to the square of the measure of the hypotenuse.0014

First of all, it is very important to remember that the Pythagorean theorem can only be used for right triangles.0021

Make sure that you have a right angle there to show that it is a right triangle.0028

And then, this side and this side are called legs.0032

When you square this leg, and you add it to the square of this leg, it is going to equal the square of the hypotenuse:0039

a² + b² = c²; and the c is representing the hypotenuse.0048

It just means "leg squared plus leg squared equals hypotenuse squared."0059

Whenever you have a missing side, if you are given two sides of a triangle,0066

and you are missing the third side, then you can always use the Pythagorean theorem--of course, as long as it is a right triangle.0071

Let's go over the converse: now, the Pythagorean theorem converse says that, if the sum of the squares0080

of the measures of two sides is equal to the square of the measure of the longest side, then the triangle is a right triangle.0087

It is like the same thing as the Pythagorean theorem; it is just the converse.0098

It is just saying, "Well, if a² + b² = c², c being the longest side, then it is a right triangle."0101

The Pythagorean theorem says that, if it is a right triangle, then a² + b² = c².0119

This one is just saying, "Well, if a² + b² = c², then it is a right triangle."0125

When can you use this?--when you are given all three sides of a triangle, and you have to determine if it is a right triangle or not.0133

If you are given all three measures of all of the sides, then see the longest one; and then, see if this works.0141

If this works, then it is a right triangle; if it doesn't work, then it is not a right triangle.0155

A Pythagorean triple is a set of three numbers that satisfies the equation a² + b² = c².0163

But they can't just be any numbers; it has to be three positive whole numbers--no fractions; no decimals; just whole numbers--0174

three whole numbers that can be the measures of the three sides of a right triangle; that is a Pythagorean triple.0186

Again, if any one of these, A, B, or C, is going to be a fraction or a decimal, then that wouldn't be a Pythagorean triple.0197

They just have to be whole numbers; and that is a triple (triple meaning three)--a group of three whole numbers.0203

One example of a Pythagorean triple is going to be 3-4-5.0214

Now, I know this because 5 has to be my hypotenuse, because that is the biggest number;0220

so let's use the Pythagorean theorem: 3² + 4² = 5².0229

3 squared is 9, plus 16, equals 25; this equals 25; so then, we know that, since this is true, 3-4-5 would be a Pythagorean triple.0239

And then, since 3-4-5 is always a Pythagorean triple, whenever you are given a right triangle,0259

and let's say one is 3, one is 5, and they are asking you for this, since you know that 3-4-5 is a Pythagorean triple,0266

x has to be 4; these are going to work together to be the three sides of a right triangle: 3-4-5.0278

The same thing: if this is 4, and this is 5, and this is the unknown, then you know that that has to be 3,0287

because it has to be these three that go together to form the Pythagorean triple: 3-4-5.0292

OK, our examples: Find the missing side.0302

From here, you can go ahead and just take out your calculator, and you can work on these problems.0308

Eric...sorry...oh, never mind; I wrote it on the bottom, but I guess I wrote it for the next one, not this one;0317

so I was wondering, "Where did it go?"...that was for #3, I think; OK, sorry.0331

So it is all good; just restart from Example 1.0340

I was wondering, "Where did my writing go?"0346

I was thinking we had to save it and reload it or something, but...never mind; I'm sorry.0348

OK, for our examples, for the first one, we are going to find the missing side.0363

We have a right triangle, so then we know that we can use the Pythagorean theorem.0368

Here, we have a leg, a leg, and then the hypotenuse is what is missing; so 11² + 14² = x².0374

Now, you can go ahead and use your calculator; take it out and work with me here.0387

11² is 121, plus 14² is 196, and that is going to be equal to x².0392

So, 121 + 196 is going to be 317 = x², and then x is going to be the square root of 317.0403

And I believe that that is a prime number; you might be able to change that to a decimal, or leave it like that.0423

If this doesn't simplify, then you leave it like that.0435

Otherwise, you can put a decimal; but that would be the answer for x.0438

The next one we are going to do is: leg squared, so 1.9², plus x², and then0445

there is a hypotenuse that is 3.2²...so the leg is what we are looking for.0461

1.9² is 3.61, and then, 3.2² is 10.24.0488

So then, x²...we are going to have to subtract those numbers, and we are going to get...it is going to be 6.63.0493

And then, x is going to equal the square root of 6.63; and you can just use your calculator to figure that out...there.0512

Example 2: Determine if the triangle with the given measures is a right triangle.0527

Here, we are going to be using the converse of the Pythagorean theorem,0532

because we were given three sides, and we have to determine if it is a right triangle.0537

It is really important, when you look at these numbers, that you have to look for the biggest one,0547

the largest number, and make that become C, or act as the hypotenuse, if it is going to be a right triangle.0552

I am going to do 4.2² + 5.5² = 6.9².0563

And 4.2² is going to be 17.64, plus 5.5² is 30.25, and then 6.9² is 47.61.0577

When we add these together, it is going to be...0604

And when you are looking to see if this is a right triangle, you don't know if this equals this yet.0606

This...we don't know if it equals this, so what we are going to do:0613

instead of just writing an equals sign and making it look like they equal each other, we can put a question mark0617

over the equals sign, because that is what we are trying to determine--if this side is equal to this side.0622

And if it is, that is what is going to determine that it is a right triangle.0628

Put a question mark over the equals sign; that way, you know that that is what you are looking for.0634

It is still a question mark; you are figuring out if it is equal.0638

The next step: we are going to add these up, and it is going to be 47.89.0644

And now, it is not really equal, but we are just going to say it is, because we have a lot of decimals here,0652

and when you round, it probably just rounded a number too closely.0663

But we are just going to say that it is about the same; it is close enough so that it could be equal.0671

We are just going to say "yes" for this one.0677

The next one: let's see, the largest one is this one right here, so I am going to do 10² + 8.5² = 12.2².0685

10² is 100, plus 8.5² is 72.25; and 12.2² is 148.84.0697

This becomes 172.25...with a question mark, and then put a question mark with that one, too.0717

So then, here we know that this does not equal this.0729

Instead of leaving the equals sign, we are going to put a line through it, and that means "not equal to."0734

So then, a² + b² is not equal to c²; therefore, this is not a right triangle.0739

If these are the measures of the three sides of a triangle, that triangle would not be a right triangle.0748

The next example: Determine if each triple is a Pythagorean triple, "triple" meaning that we have a group of three numbers.0757

And are those three numbers that make up the sides of a right triangle?0766

You pretty much do the same thing here: you use the Pythagorean theorem converse.0774

8² + 10² is equal to 12²; so, 64 + 100 = 144?0781

164 does not 144, so this one is "no."0794

The next one: 5² + 12² = 13²; 25 + 144 = 169.0803

This is going to be 169 = 169, so this one is "yes," because this side, a², plus b², equals c².0820

This one is "yes"; this is a Pythagorean triple.0835

And then, the next one: 8² plus the largest one, 15²...0839

now, for these examples that we are doing here, the largest number just happens to be the last one;0846

but in your book, it is not always going to be like that; so just make sure that you look at all of the numbers,0853

and see which one is the largest one; and that one has to be c.0860

You can't have 15² + 17² equaling 8², so just keep that in mind.0866

Here, 8² is 64, plus 225, is equal to 289.0875

And then, when we add this up, it is going to be 289, which equals 289; so here we have another Pythagorean triple.0890

Pythagorean triple, Pythagorean triple...5, 12, 13: this one, 5, 12, 13, is actually pretty common.0901

The 3-4-5 and the 5-12-13--we will just keep that in mind.0909

That way, when you are given problems, you can just use these three; you know that these three go together.0918

If one of these is missing, then you know that these have to go together.0925

8-15-17 is also another Pythagorean triple.0932

The last example: Given the coordinates of the three vertices of a triangle, determine if the triangle is a right triangle, without using slope.0937

Now, it says "without using slope" because, if you use slope, then you can see that two sides0950

can have the negative reciprocal slope, and therefore be perpendicular, which would make a right angle,0964

which then would make this triangle a right triangle.0971

So, instead of doing that, without using slope, here let's say that this is a, this is b, and this is c.0976

We just have a triangle, and in order to use the Pythagorean theorem, you have to have the measure of the length of the side.0988

We don't have any lengths; we just have the points; so we have to find the distance between the two points to find the length.1005

We have to use the distance formula; this would be x² + (y₂ - y₁)².1014

And then, that way, we can find the length of each side, and then we will use those sides in the Pythagorean theorem.1028

Here, let's see: AB, first, is going to be the square root of...(2 - -1), which is 2 + 1...plus 3 - 0 squared;1040

that is going to be 3² + 3², which is 9 + 9, is 18, so √18.1062

Now, I know that this can simplify, but I am just going to leave it for now.1075

We will leave them all, just like that; let's see, BC...it doesn't matter which order you do it in;1080

you can do AC first, or...it doesn't matter; BC would be -1 - -4, squared, plus 0 - 2, squared.1090

This is going to be -1 + 4, which is 3, squared, plus -2, squared; and that is going to be √(9 + 4), so √13.1109

And then, AC is 2 - -4, squared, plus 3 - 2, squared; and then, this becomes plus; so 6 squared is 36, plus 1 squared;1127

so it is going to be the square root of 37.1157

So, I have the square root of 18, the square root of 13, and the square root of 37.1163

Now, these are the measures of my sides.1169

Now, if I need to plug this into the Pythagorean theorem, to see if it is going to be a right triangle,1175

I have to make sure that my longest will be c.1182

Now, since they are all just the square root in radical form, I can just see that this one is the largest number.1185

This is going to be my c; so when I plug it into the Pythagorean theorem, I am going to make it1194

the square root of (I'll use a different color) 18, squared, plus the square root of 13,1201

that side squared, equals the square root of 37, squared.1217

Square root and square are opposites; they cancel each other out, so the square root of 18 squared will just be 18.1227

This whole thing and this whole thing will cancel each other out, just like if I multiply a number by 2,1238

and then divide it by 2, it just becomes that number.1246

If I do, let's say, 2 times x, divided by 2, what happens to these two numbers?1249

They cancel out, and I just get x.1258

If I add 2 and then subtract 2, it is the same thing; they are opposites, so they can cancel each other out.1261

The same thing happens here: square root and squared are opposites, so they cancel each other out.1267

So, it just becomes 18, plus...the square root of 13 squared becomes 13; that equals...this is...I wrote 32, but it is 37.1272

37...question marks...this is 31; 31 does not equal 37, so then, these three would not be the vertices of a right triangle.1288

These are the coordinates of the vertices of a triangle, but it is not of a right triangle,1314

because we plugged all of the measures of the sides into the Pythagorean theorem, and it doesn't work.1320

a² + b² does not equal c², so this one is "no"; it is not a right triangle.1327

That is it for this lesson; thank you for watching Educator.com.1338