WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this next lesson, we are going to go over special right triangles.
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Now, we are still on right triangles; so make sure that all of these are only being used for right triangles.
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That includes the Pythagorean theorem; that includes the altitude to find the geometric mean, and this one, which is special right triangles.
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Now, we have two types of special right triangles.
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Now, what makes them special: it is a right triangle, but it is special because of the type of right triangle that it is.
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If it is a 45-45-90 degree triangle, meaning that the angle is 45, the other angle is 45, and then, of course,
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we have the 90-degree right angle, then there is a shortcut to finding the other sides of the triangle.
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That is why it is special--because there is a shortcut.
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Again, there are two of them: if you have any other type of triangle, besides these two special right triangles,
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then you are going to have to use one of the other methods, depending on what you are trying to find.
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For a 45-45-90 degree triangle, we don't have to use the Pythagorean theorem;
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we don't have to use geometric mean and some of the other concepts that we are going to go over in the next lesson.
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For the 45-45-90 triangle, we know that the hypotenuse is going to be √2 times [as long as the leg].
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Now, we know that a triangle has three sides and three angles.
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For each angle, there is a side that is corresponding with it; it is always the angle and the side opposite that angle that kind of pair up.
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They are kind of like a couple; they pair up, so they are related.
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We have...if this is triangle ABC, then I can say that the side opposite angle A...this...I can name that side a (lowercase a for the sides).
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And then, here, this side AC is going to be corresponding with this angle B; so it is as if that is b, because this angle and this side go together.
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And then, for this angle C and the hypotenuse, they are kind of related, too.
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Keep that in mind--that is very important to remember: the angle and the side opposite go together.
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In a 45-45-90 triangle, how do I know that this is a 45-45-90 degree triangle?
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It is only given that this is 45 and this is 90, but I know that all three angles of a triangle have to add up to 180.
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So, if this is given to me as 45, and this is given, then I know that the third angle has to be 45, as well.
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45-45-90: the side opposite the 45-degree angle, we are going to name as n; so then, these two are going to go together.
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So then, this one is going to be n; now, I am going to erase this, so that you don't get it confused with all of these variables.
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The side opposite the 45 is n; well, if the side opposite this 45 is n, then the side opposite this 45 also has to be n,
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because remember: this is an isosceles triangle, and because of the base angles theorem,
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if these angles are the same, then those sides opposite have to be the same.
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If this is n, then this has to also be n.
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And then, the hypotenuse, then, the side opposite the 90-degree angle, is going to be √2 times that side.
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So, if this is n, then this is going to be √2n, or let me write n√2, n times √2.
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Just keep this in mind: 45-45-90 is going to be n, n, n√2.
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Let's say that this CB is, let's say, 4; then this side right here is going to be 4, and AB is going to be 4√2,
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because it is √2 times n; n is 4; 4 times √2 is 4√2.
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That is the shortcut for a 45-45-90 degree triangle: n, n, n√2.
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Here, they give you one of the sides, and you have to find n.
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Now, for this one, this one is a little bit different; why?--because we know that, if this is n, what is this? n.
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What about the side opposite the 90?--45-45-90--it is going to be n, n (for the side opposite this 45),
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and then, for this one, it is going to be n√2; the side opposite the 45 is n; this is also 45,
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so that is going to be n; what about the side opposite the 90?--it is going to be n√2.
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We don't know n; we want to find n.
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They give us this one right here, so we know that this is 10.
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What I can do, since I know that this is equal to n√2, which is 10: I can just make that into an equation.
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n√2, this side right here, is 10; I can solve for n.
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So then, how do I do this? I would have to divide by √2, so n is equal to 10/√2.
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Remember: I can't have a radical in my denominator, so I have to rationalize it.
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I have to multiply this by √2/√2, because √2/√2 makes one; anything over itself is 1.
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So, when I multiply this out, it is going to be 10√2 over...what is √2 times √2? 2.
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And then, this will become 5√2; so n is equal to 5√2.
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That means that this is 5√2, and this is 5√2.
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Again, just make sure that you understand this; this is n; this is n; then, this would be n√2.
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Whatever side they give you, as long as they give you one side--you can find the remaining sides using the shortcut, using this.
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But you just have to make sure that you are going to make that measure equal to one of these, depending on what side they give you.
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If they give you one of the sides opposite the 45, then that is n.
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You are going to make that equal to n.
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If they give you the hypotenuse, you are going to make that equal to n√2, and then you are going to solve for n by making it equal to each other.
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So, on to the next special right triangle: that is a 30-60-90 triangle.
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If you have a triangle whose angle measures are 30, 60, and 90, then we can also use another shortcut.
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Here, they don't have to give you that this is 60, as long as they give you that one of them is a right triangle, and one of them...
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this is either 60, or this is 30; then you know that it is a 30-60-90 triangle,
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because, if you are going to just subtract it from 180, you are going to get that remaining angle measure.
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And it is going to become either 30 or 60.
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So then, a 30-60-90 triangle: the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.
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The shorter leg is the side opposite the 30-degree angle: 30, 60, 90.
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This is a little bit trickier; the side opposite the 30...we are going to name that n.
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Now, here, this 60-degree one, the longer leg, is going to be n√3.
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Now, be careful here, because the 2n, the one that is twice as long as the shorter leg, is the hypotenuse, 2n.
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This is probably where students make the biggest mistake, because 60 is two times 30,
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so they automatically assume that, if this is n, then this has to be 2n.
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That is not the case; this is n, n√3, and then this one, the one opposite the 90, the hypotenuse, is 2 times n.
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So, be very careful with that: n, n√3, and 2n.
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Let's write the 45-45-90; this one is n, n, n√2; this is n√3 for the 60 one.
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So, be careful; this is n√2; this is n√3; that is also another common mistake.
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Instead of writing n√3, they write n√2; or with this one, instead of writing n√2, they write 3; so just be careful with that.
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Let's say that this one right here is 4; then, the one opposite the 60 would be n√3, so that would be 4 times √3, 4√3.
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And the hypotenuse is going to be 2 times that side, n; so 2 times n is 8.
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So, if it is 4 here, then it is going to be 4√3 here, and it is going to be 8 there.
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Let's do one example of this: again, let's write it out: a 30-60-90 triangle is going to be n; this one is not 2n, but n√3; this one is going to be 2n.
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Here, this is n; this one...oh, this is 30 degrees; this is 60 degrees...the one opposite the 30 is n, which is what they want you to find.
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The one opposite the 60 is 12; remember, again, as long as they give you the measure of one side,
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you can find the remaining sides, because we have a shortcut here.
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Here, this one opposite 60 is 12; that means that this is 12, right here.
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And then, the hypotenuse we don't know; all we know is that that is that, and then we know that 12 is equal to...
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this is n; this is n√3; this is 2n; we know that n√3 is equal to 12.
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So, all you have to do is make them equal to each other: n√3 is equal to 12.
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To solve for n, divide by √3; n is equal to 12/√3.
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Again, we have to simplify this by rationalizing the denominator.
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√3/√3 becomes 1; we have to do that, because we need to get rid of this radical; it will become 12√3/3, so n is 4√3.
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n is 4√3; that means that this one here is 4√3.
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And then, what about the hypotenuse? It is 2 times that number, so it is 2 times 4√3, which is 8√3.
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So again, whatever side they give you, just make it equal to that shortcut; and then, you just solve out for n.
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Let's do a few examples: Find the values of x and y.
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Here is a 45-45-90 degree triangle; that means that we can use our shortcut.
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They gave us one side; the 45-45-90 one is n, n, n√2.
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What do they give us? They give us the one opposite the 45, so they give us this one right here.
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This one is 15; so what are the other sides?
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This one here, opposite the 45, is also 15.
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And then, how about this one? This would be 15 times √2, which is 15√2.
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So again, this one is given, the side opposite the 45.
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Then, we can find our other two sides, because if 15 is n, then what is n here? 15 is n.
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Then, n√2 would be 15√2.
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Here we have a 30-60-90 degree triangle; 30 is n; this one is not 2n, but n√3; and this one is 2n.
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Which one is given? Let's see, this one, the hypotenuse, is given, and that is the one opposite the right angle, right here.
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Here, this is going to be 22; that means I can just make that equal to each other.
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Since this is 2n, 2n is equal to 22; so n is 11.
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Which one is n? Isn't that here, the one opposite the 30?
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So, this is 11, and then this one is going to be 11√3.
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The side opposite 30, x, is 11; and then, y is the side opposite 60, the long leg, so it is 11√3.
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Let's write out x and y for this one, too.
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If this side is 15, then x is 15; what is y? x√2...so it is 15√2.
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The next one, Example 2: Find the values of x and y.
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OK, so here, this is actually a square; and this is a diagonal of the square, so that makes this angle a 45-degree angle.
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Automatically, I know that x is 45 degrees, which means that this would also be 45; here is a right angle; here is 45.
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This right here now becomes a 45-45-90 degree triangle.
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45, 45, 90: the side opposite the 45 is n, so this is going to be n; the side opposite 90 is going to be n√2.
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Now, what are we given? We are given the hypotenuse, 16, so that is this right here; the side opposite the 90 is 16.
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That means that I am going to make this equal to that, so n√2 = 16.
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Divide by √2; n is equal to 16/√2; multiply this by √2/√2; 16√2/2--that makes n 8√2, which is, in this case, y.
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In this problem, it is y, not n.
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So then, here, y is equal to 8√2; those are my answers.
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The next one: let's see, 30-60-90...how do I know that?--because look at that triangle right there: 30, 60, 90.
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Let's see, they want to know this right here and this whole side right here.
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If 30 is n, 60 will be n√3, and 90 will be 2n.
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From this triangle, this right here is also 60, which would make this 30.
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Now, it is the same thing to find this right here, to use this triangle, or to use this triangle, because it is still 30-60-90.
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Here is the hypotenuse; here is x, what we are looking for; but what is given?
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Look at what is given: they have to give you at least one side, so in this problem, they give you the side opposite the 60, the long leg.
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This one is 10.5; so we are going to have to solve that out and make them equal to each other: 10.5,
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then divide...n is equal to 10.5 divided by √3.
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Now, this is decimals, so you want to use your calculator.
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n is 10.5√3, over 3; and again, use your calculator.
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Or if you want to leave it as a fraction, then you can just change this 10.5; 10.5, we know, is 10 and 1/2.
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If you change that to an improper fraction, it becomes 21/2; so then, this can be the same thing as 21/2 √3/3.
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Now, this is also this, which is 21√3/6.
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All I did was just to change 10.5 into a fraction, which became 21/2.
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And then, I multiplied it to √3, so it became 21√3/2.
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And then, you just divide it by 3; and when you divide, it just becomes 21√3/6.
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You can leave it like that, as long as you don't have a radical in the denominator,
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and as long as these numbers don't simplify; then that would be the answer.
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Otherwise, you can just change it to decimals; because this is given in decimals, you can go ahead and use your calculator and find the decimal of that.
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I don't have a calculator that I can use to give you the decimal answer, so you would have to just punch in...
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you could do 21 times √3 over 6, divided by 6, or do 21 divided by 6, and multiply that by √3.
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I am just going to say that this n, which is the side opposite here, is 21√3/6.
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Now, they are not asking for this side right here; they are asking for this side, which is the side opposite the 90, the hypotenuse.
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So, I need to multiply that by 2; 2 times this whole thing is going to be (I'll do it right here) 21√3/6.
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Now, I can just go ahead and simplify this, or I can multiply it out; it becomes 42√3/6, and then you have to simplify.
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Or just change this to a 3; so it becomes 21√3/3, which would then become 7√3.
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x is 7√3, and then y is also the same answer.
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Now, if this right here was 21√3/6--this was the answer to this--why is it just 2 times that amount?
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Now, if you look here, this is 60, this is 60, and this is 60.
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That makes this an equilateral triangle--equiangular, and therefore equilateral.
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Whatever this side becomes, this side is, also.
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Now, you can also think of it as these two triangles being the same; so if this is 21√3/6, then this is 21...
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this actually simplifies, so this becomes 2, and then this becomes 7; so it becomes 7√3/2.
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This is also 7√3/2; so if you just add them together (or you can just do 7√3/2 times 2, and that is the same thing), y is 7√3, the same answer.
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The third example: The perimeter of an equilateral triangle is 27 inches; find the length of an altitude of the triangle.
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We have an equilateral triangle; the perimeter is 27 inches.
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So, if it is equilateral, that means I know that this side, this side, and this side are all the same,
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which would make each side 9, because it is 27 divided by 3.
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Find the length of an altitude of the triangle.
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If it is an equilateral triangle, then it is also equiangular, which means that this is 60, 60, and 60,
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because all three angles of a triangle have to add up to 180, so it is 180 divided by the three angles.
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I need to find an altitude; an altitude is that right there.
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If this is 60 and this is 90, then this has to be 30; that means we are using a 30-60-90 triangle.
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Then, here is 30; that is n; this is n√3; here, this is 2n.
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What is given to us? The hypotenuse, the side opposite the 90--this one is 9.
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So then, I just need to make those two equal to each other; divide by 2; n is...I can leave it as 9/2, or 4.5.
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And what do they want us to find?
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They want us to find this right here, the altitude.
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What did we find? We found the side opposite the 30; this one is 4.5.
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How do you find the one opposite the 60?
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It is n times √3, so we have to do x = 4.5 times √3, so it is 4.5√3; that is the altitude.
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If you want to leave it as a fraction, you can say that it is 9/2, times √3; it is going to be 9√3/2--that is also the answer.
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The next one: the length of a diagonal of a rectangle is 15 centimeters long and intersects the vertex to make a 60-degree angle.
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Find the perimeter of the rectangle.
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The rectangle...the length of a diagonal...I need a diagonal; this is 15; these are our right angles.
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This intersects the vertex to make a 60-degree angle.
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This whole thing is 90; that means that, if one of these is 60, the other one has to be 30.
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So, I want to make the bigger one 60; this is going to be 30.
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Find the perimeter of the rectangle: if this is 30, then this is 60; it has to be a 30-60-90 degree triangle.
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30 + 60 + 90 has to be 180; find the perimeter.
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OK, if I am going to find the perimeter, that means that I have to find the measure of all of my sides.
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I have to find this side, and I have to find this side.
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With 30-60-90, this is n, n√3, 2n.
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They give us the side opposite the 90, which is this, 15; make them equal to each other.
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n = 15/2; that is this one right here, 15/2, the side opposite 30.
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What is this? It is 15/2, and then what is the one opposite the 60?
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It is n√3, so it is 15/2 times √3, or 15√3/2.
00:32:44.000 --> 00:32:51.900
To find the perimeter, I have to add up all of the sides, so it is going to be this side times 2,
00:32:51.900 --> 00:33:12.400
so the perimeter is going to be 2(15√3/2) + 2 times this side, 15/2...
00:33:12.400 --> 00:33:21.800
Again, perimeter is just this plus this; it is 2 times this side, plus 2 times this.
00:33:21.800 --> 00:33:31.700
So then, this becomes (I'll just cancel that out) 15√3 + 15.
00:33:31.700 --> 00:33:38.900
And I cannot add these two up, because this one has √3, so this would just be the answer.
00:33:38.900 --> 00:33:51.500
I'll just write it out here: the perimeter equals 15√3 + 15, and that would be the answer.
00:33:51.500 --> 00:34:03.000
Next, Example 4: The regular hexagon is made up of 6 equilateral triangles; find AC.
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A regular hexagon..."regular" means that it is equilateral and that it is equiangular.
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Here is a hexagon, because it has six sides; and "regular"--each of these sides is congruent.
00:34:21.000 --> 00:34:38.100
I know that, if it is regular, then that means that it is made up of six congruent triangles.
00:34:38.100 --> 00:34:55.800
Each of these triangles is congruent; that means that this side is going to be 12; these angles are going to be 60,
00:34:55.800 --> 00:35:01.700
because again, "equilateral triangles" would also mean that they are equiangular,
00:35:01.700 --> 00:35:10.500
and that means that all of the angles have to be 60 (because together it has to be 180, and then you just divide it by 3).
00:35:10.500 --> 00:35:20.100
We have to find AC; here, this is perpendicular...so then, if we look at this triangle right here,
00:35:20.100 --> 00:35:32.300
this one with this one and this, this is 60; what would that make this?
00:35:32.300 --> 00:35:39.600
This would be 30, because it is these two parts; so here, we have a 30-60-90 triangle.
00:35:39.600 --> 00:35:49.200
Now, if I find this right here, then I can just multiply it by 2 to get AC.
00:35:49.200 --> 00:36:05.100
I know that this side right here is 12; I am going to be using a 30-60-90 triangle, n, n√3, and 2n.
00:36:05.100 --> 00:36:23.700
From here, look at that triangle right there; what side is given to me?--the hypotenuse, which is this: 2n = 12, so n is 6.
00:36:23.700 --> 00:36:29.200
That is this right here; so this is 6; what is this right here, then?
00:36:29.200 --> 00:36:34.700
That would be n√3, which is 6√3.
00:36:34.700 --> 00:36:41.700
I need to find AC; well, if this is 6√3, that means that this has to be 6√3, also.
00:36:41.700 --> 00:37:02.100
So, AC then would be 2 times 6√3, so AC is 12√3.
00:37:02.100 --> 00:37:09.400
Just to go over what we just did: this hexagon is made up of 6 equilateral triangles,
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so I just split it up into 6 triangles; I am going to focus on this triangle right here, and then half of that; that way, I can find this right here.
00:37:22.000 --> 00:37:27.300
And then, since I know that I have a 30-60-90 triangle, I can use my shortcut.
00:37:27.300 --> 00:37:38.500
My hypotenuse is 12, so I made 12 equal to 2n, solved for n, and then found this side, which is 6,
00:37:38.500 --> 00:37:43.100
and then this side, which is 6√3; this is what I need.
00:37:43.100 --> 00:37:53.700
I multiply that by 2, so I get 12√3; so AC, which is what they want me to find, is 12√3.
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That is it for this lesson; thank you for watching Educator.com.