WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This next lesson, we are going to go over the Pythagorean theorem.
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The Pythagorean theorem says that, in a right triangle, the sum of the squares of the measures of the legs
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is equal to the square of the measure of the hypotenuse.
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First of all, it is very important to remember that the Pythagorean theorem can only be used for right triangles.
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Make sure that you have a right angle there to show that it is a right triangle.
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And then, this side and this side are called legs.
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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:
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a² + b² = c²; and the c is representing the hypotenuse.
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It just means "leg squared plus leg squared equals hypotenuse squared."
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Whenever you have a missing side, if you are given two sides of a triangle,
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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.
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Let's go over the converse: now, the Pythagorean theorem converse says that, if the sum of the squares
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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.
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It is like the same thing as the Pythagorean theorem; it is just the converse.
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It is just saying, "Well, if a² + b² = c², c being the longest side, then it is a right triangle."
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The Pythagorean theorem says that, if it is a right triangle, then a² + b² = c².
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This one is just saying, "Well, if a² + b² = c², then it is a right triangle."
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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.
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If you are given all three measures of all of the sides, then see the longest one; and then, see if this works.
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If this works, then it is a right triangle; if it doesn't work, then it is not a right triangle.
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A Pythagorean triple is a set of three numbers that satisfies the equation a² + b² = c².
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But they can't just be any numbers; it has to be three positive whole numbers--no fractions; no decimals; just whole numbers--
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three whole numbers that can be the measures of the three sides of a right triangle; that is a Pythagorean triple.
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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.
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They just have to be whole numbers; and that is a triple (triple meaning three)--a group of three whole numbers.
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One example of a Pythagorean triple is going to be 3-4-5.
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Now, I know this because 5 has to be my hypotenuse, because that is the biggest number;
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so let's use the Pythagorean theorem: 3² + 4² = 5².
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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.
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And then, since 3-4-5 is always a Pythagorean triple, whenever you are given a right triangle,
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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,
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x has to be 4; these are going to work together to be the three sides of a right triangle: 3-4-5.
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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,
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because it has to be these three that go together to form the Pythagorean triple: 3-4-5.
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OK, our examples: Find the missing side.
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From here, you can go ahead and just take out your calculator, and you can work on these problems.
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Eric...sorry...oh, never mind; I wrote it on the bottom, but I guess I wrote it for the next one, not this one;
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so I was wondering, "Where did it go?"...that was for #3, I think; OK, sorry.
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So it is all good; just restart from Example 1.
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I was wondering, "Where did my writing go?"
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I was thinking we had to save it and reload it or something, but...never mind; I'm sorry.
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OK, for our examples, for the first one, we are going to find the missing side.
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We have a right triangle, so then we know that we can use the Pythagorean theorem.
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Here, we have a leg, a leg, and then the hypotenuse is what is missing; so 11² + 14² = x².
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Now, you can go ahead and use your calculator; take it out and work with me here.
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11² is 121, plus 14² is 196, and that is going to be equal to x².
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So, 121 + 196 is going to be 317 = x², and then x is going to be the square root of 317.
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And I believe that that is a prime number; you might be able to change that to a decimal, or leave it like that.
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If this doesn't simplify, then you leave it like that.
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Otherwise, you can put a decimal; but that would be the answer for x.
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The next one we are going to do is: leg squared, so 1.9², plus x², and then
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there is a hypotenuse that is 3.2²...so the leg is what we are looking for.
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1.9² is 3.61, and then, 3.2² is 10.24.
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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.
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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.
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Example 2: Determine if the triangle with the given measures is a right triangle.
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Here, we are going to be using the converse of the Pythagorean theorem,
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because we were given three sides, and we have to determine if it is a right triangle.
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It is really important, when you look at these numbers, that you have to look for the biggest one,
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the largest number, and make that become C, or act as the hypotenuse, if it is going to be a right triangle.
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I am going to do 4.2² + 5.5² = 6.9².
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And 4.2² is going to be 17.64, plus 5.5² is 30.25, and then 6.9² is 47.61.
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When we add these together, it is going to be...
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And when you are looking to see if this is a right triangle, you don't know if this equals this yet.
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This...we don't know if it equals this, so what we are going to do:
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instead of just writing an equals sign and making it look like they equal each other, we can put a question mark
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over the equals sign, because that is what we are trying to determine--if this side is equal to this side.
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And if it is, that is what is going to determine that it is a right triangle.
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Put a question mark over the equals sign; that way, you know that that is what you are looking for.
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It is still a question mark; you are figuring out if it is equal.
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The next step: we are going to add these up, and it is going to be 47.89.
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And now, it is not really equal, but we are just going to say it is, because we have a lot of decimals here,
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and when you round, it probably just rounded a number too closely.
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But we are just going to say that it is about the same; it is close enough so that it could be equal.
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We are just going to say "yes" for this one.
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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.
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10² is 100, plus 8.5² is 72.25; and 12.2² is 148.84.
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This becomes 172.25...with a question mark, and then put a question mark with that one, too.
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So then, here we know that this does not equal this.
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Instead of leaving the equals sign, we are going to put a line through it, and that means "not equal to."
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So then, a² + b² is not equal to c²; therefore, this is not a right triangle.
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If these are the measures of the three sides of a triangle, that triangle would not be a right triangle.
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The next example: Determine if each triple is a Pythagorean triple, "triple" meaning that we have a group of three numbers.
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And are those three numbers that make up the sides of a right triangle?
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You pretty much do the same thing here: you use the Pythagorean theorem converse.
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8² + 10² is equal to 12²; so, 64 + 100 = 144?
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164 does not 144, so this one is "no."
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The next one: 5² + 12² = 13²; 25 + 144 = 169.
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This is going to be 169 = 169, so this one is "yes," because this side, a², plus b², equals c².
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This one is "yes"; this is a Pythagorean triple.
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And then, the next one: 8² plus the largest one, 15²...
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now, for these examples that we are doing here, the largest number just happens to be the last one;
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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,
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and see which one is the largest one; and that one has to be c.
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You can't have 15² + 17² equaling 8², so just keep that in mind.
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Here, 8² is 64, plus 225, is equal to 289.
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And then, when we add this up, it is going to be 289, which equals 289; so here we have another Pythagorean triple.
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Pythagorean triple, Pythagorean triple...5, 12, 13: this one, 5, 12, 13, is actually pretty common.
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The 3-4-5 and the 5-12-13--we will just keep that in mind.
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That way, when you are given problems, you can just use these three; you know that these three go together.
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If one of these is missing, then you know that these have to go together.
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8-15-17 is also another Pythagorean triple.
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The last example: Given the coordinates of the three vertices of a triangle, determine if the triangle is a right triangle, without using slope.
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Now, it says "without using slope" because, if you use slope, then you can see that two sides
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can have the negative reciprocal slope, and therefore be perpendicular, which would make a right angle,
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which then would make this triangle a right triangle.
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So, instead of doing that, without using slope, here let's say that this is a, this is b, and this is c.
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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.
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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.
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We have to use the distance formula; this would be x² + (y₂ - y₁)².
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And then, that way, we can find the length of each side, and then we will use those sides in the Pythagorean theorem.
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Here, let's see: AB, first, is going to be the square root of...(2 - -1), which is 2 + 1...plus 3 - 0 squared;
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that is going to be 3² + 3², which is 9 + 9, is 18, so √18.
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Now, I know that this can simplify, but I am just going to leave it for now.
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We will leave them all, just like that; let's see, BC...it doesn't matter which order you do it in;
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you can do AC first, or...it doesn't matter; BC would be -1 - -4, squared, plus 0 - 2, squared.
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This is going to be -1 + 4, which is 3, squared, plus -2, squared; and that is going to be √(9 + 4), so √13.
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And then, AC is 2 - -4, squared, plus 3 - 2, squared; and then, this becomes plus; so 6 squared is 36, plus 1 squared;
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so it is going to be the square root of 37.
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So, I have the square root of 18, the square root of 13, and the square root of 37.
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Now, these are the measures of my sides.
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Now, if I need to plug this into the Pythagorean theorem, to see if it is going to be a right triangle,
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I have to make sure that my longest will be c.
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Now, since they are all just the square root in radical form, I can just see that this one is the largest number.
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This is going to be my c; so when I plug it into the Pythagorean theorem, I am going to make it
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the square root of (I'll use a different color) 18, squared, plus the square root of 13,
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that side squared, equals the square root of 37, squared.
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Square root and square are opposites; they cancel each other out, so the square root of 18 squared will just be 18.
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This whole thing and this whole thing will cancel each other out, just like if I multiply a number by 2,
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and then divide it by 2, it just becomes that number.
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If I do, let's say, 2 times x, divided by 2, what happens to these two numbers?
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They cancel out, and I just get x.
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If I add 2 and then subtract 2, it is the same thing; they are opposites, so they can cancel each other out.
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The same thing happens here: square root and squared are opposites, so they cancel each other out.
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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.
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37...question marks...this is 31; 31 does not equal 37, so then, these three would not be the vertices of a right triangle.
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These are the coordinates of the vertices of a triangle, but it is not of a right triangle,
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because we plugged all of the measures of the sides into the Pythagorean theorem, and it doesn't work.
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a² + b² does not equal c², so this one is "no"; it is not a right triangle.
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That is it for this lesson; thank you for watching Educator.com.