WEBVTT
 
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Let's review some information about triangles from geometry. 

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Recall that the sum of the angle measures in any triangle is 180 degrees.

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You are going to use that quite often.

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The other thing we can do is classify triangles

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based on either their angle measures or the lengths of their sides.

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Let's look at some definitions.

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An acute triangle is a triangle where all the angles are less than 90 degrees. 

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Remember that 90 degrees is the same thing as pi over 2 radians.

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A right triangle is a triangle that has exacly one right angle

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or 90 degrees or pi over 2,

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and an obtuse triangle is a triangle that's got one angle

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that's greater than 90 degrees or pi over 2 radians.

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Now when we classify them based on the lengths of the sides

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we have a scalene triangle, that's one where no sides are congruent,

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isosceles triangle, two sides are congruent, 

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and equilateral triangle, all three sides are congruent.

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So let's look at an example where we classify triangles

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based on their angle measures and their sides. 

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So let's look at this first example. I can see from the tick marks

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that all three sides are different lengths, so I know it's scalene.

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And then I see one angle measure here that's 2 pi over 3.

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Now you have to know that 2 pi over 3 is bigger than pi over 2.

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So this angle measure is obtuse. So this triangle would be obtuse and scalene.

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All right, let's look at the next example. Here we see a right triangle.

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Remember this symbol means a right angle. We see two sides with one tick mark,

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so we know those two sides are equal and this side is different. 

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So we've got a right triangle. Two sides are equal, so it's isosceles.

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So this would be a right, isosceles triangle.

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All right, let's look at the last example. 

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Here again I see two sides that are congruent, so I know its isosceles. 

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This side is different by the two tick marks, and then I see one angle

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measure that's 70 degrees. One thing that you need to remember from geometry

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is that when you have two sides equal, isosceles triangle, you've got the 

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base angles that are equal. Those are the angles opposite 

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these congruent sides. So these two angles both have to be equal.

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So since this angle is 70, then this angle is 70. And then of course

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I remember that the sum of the angle measures in a triangle is 180.

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So that gives me 140. This top angle is 40, so I can see that all the angle

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measures are less than 90. So this would be an acute, isosceles triangle. 

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The Pythagorean Theorem is one of the most widely recognized theorems 

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in all of mathematics. What it states is that the sum of the squares

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of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

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So here I've got a right triangle drawn. We are going to let the legs be represented with a and b.

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c is the side opposite the right angle. It's the hypotenuse.

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So this is a leg. This is a leg. 

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And this is the hypotenuse. So what the Pythagorean Theorem says

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is the sum of the squares of the legs is equal to the square of the hypotenuse. 

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And we can use the Pythagorean Theorem to find a missing side of a right triangle. 

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You're going to do that a lot and use it a lot. Let's look at some examples.

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In this first example, we know the lengths of the legs, and we are looking for the hypotenuse.

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So we can use this Pythagorean Theorem. We know that 7 squared plus 24 squared equals c squared. 

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We can use our calculator. I know that 7 squared is 49. 24 squared is 576.

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And let's see, I'm going to write c squared on the left. 49 plus 576 

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is 625. Now of course there are two numbers that when you square you get 625, but since

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we're looking for the length of a side, we will always use the positive number.

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The square root of 625 is 25, so the hypotenuse in this first example is 25.

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Let's look at the next example. Again we've got the lengths of the legs,

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and we are looking for the hypotenuse. So we can use the Pythagorean Theorem.

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I know that c squared is equal to 6 squared plus 5 squared. 6 squared is 36. 5 squared is 25.

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Add that together, and you get 61. Remember we are only going to use the positive 

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because it's the length of a side, so c in this example is the square root of 61. 

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Now you're always going to make sure that you simplify radicals if you can. 

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In this example, 61, you can't simplify, so you'll leave it just like this. 

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Let's look at the last example. Here I'm looking for a leg. 

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I know one leg is the square root of 2, and I know the hypotenuse is the square root of 11. 

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So using the Pythagorean theorem we have b squared plus the square root of 2 squared 

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equals the square root of 11 squared. OK, so that gives us b squared,

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remember when you square a square root you get the number under the radical, 

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so I have 2 and then the square root of 11 squared is 11. 

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Now we'll subtract 2 from both sides and get 9, so b in this case is 3.

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So you're going to use the Pythagorean theorem many, many times to find a missing side of a right triangle. 

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It's important that you know the Pythagorean theorem, and remember it only applies to a right triangle. 

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Let's take a look at similar triangles. 

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Similar triangles are triangles that have the same shape, but not necessarily the same size.

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There are two properties of similar triangles. Number one, the corresponding angles have the same measure, 

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and the second, the ratios of the corresponding sides are equal. 

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In other words, the corresponding sides are proportional.

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Let's look at this example. I've got two similar triangles. 

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I can see that I have angle measures that have the same measure by the tick marks.

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A has the same measure as D,

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B has the same measure as E,

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and C has the same measure as F.

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We also know that the ratios of the corresponding sides are equal.

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So corresponding sides are the sides that are opposite the congruent angles,

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the angles that have the same measure.

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So a corresponds to d so that ratio is the same as the 

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ratio of b which corresponds to e 

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which is the same as the ratio of c which corresponds to f.

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All of these ratios will be equal in similar triangles.

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Now we talked about being proportional. Let's think about this.

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We have something called the proportionality constant. 

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We use the letter k, 

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where k is greater than or equal to 1.

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So the proportionality constant is equal to the length 

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of a side of the larger triangle divided by the length

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of the corresponding side of the smaller triangle.

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If the triangles are the same size, then k is 1.

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So let's look at this example and find the proportionality constant k.

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Well we can see that we've got angle measures that are the same, again by the tick marks.

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A and D have the same measure, B and E have the same measure, 

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C and F have the same measure.

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So these sides correspond because they're opposite 

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congruent angles that have the same measure.

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So in order to find that proportionality constant we want to know 

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the larger side which is 8

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divided by the corresponding side in the smaller triangle, 4.

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Well that happens to be 2.

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So in this example our proportionality constant is 2.

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And we can use proportionality constants to find missing sides of similar triangles.

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Let's look at some examples of that.

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Let's look at some examples of finding missing sides 

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of similar right triangles by finding the proportionality constant, k.

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In this first right triangle we see we have two legs.

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We know the lengths of the legs.

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We can use the Pythagorean theorem to find the hypotenuse.

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So let's do that. So AB squared is equal to 56 squared plus 33 squared.

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So 56 squared is 3136, 

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and 33 squared is 1089. Add those together.

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We get AB squared is 4225. We'll take the square root of that,

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and we get AB is 65. So now we've found the missing side of the first triangle.

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Now we know the triangles are similar.

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So this is 65.

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We can find that proportionality constant by taking the only side I know in the second triangle,

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the larger triangle, 260. Well this side, this hypotenuse, corresponds to 65, 

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the hypotenuse in the first triangle, and 260 divided by 65 is 4.

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So my proportionality constant is 4. So that's going to make finding XZ and YZ easy.

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All I have to do to find XZ is multiply its corresponding side in the smaller triangle, 56, by 4.

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56 times 4, let's just do that on our calculator.

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56 times 4 is 224. So that's the length of XZ, and then to find YZ you're going to find

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its corresponding side is 33. So we're going to multiply 33 by 4, 

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and we get 132. So those are the missing sides of that first example.

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Let's look at another example.

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In the second example, the triangles are similar again,

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but notice that they're not oriented in the same direction.

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So what you might want to do to help you is let's redraw this triangle here.

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We can draw it. Let's see. This is R...

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So now the triangles are oriented in the same direction.

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Let's see which side we knew. RS is 1.

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So now, again we have similar triangles.

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In the second right triangle, we know a leg and the hypotenuse,

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so we can use the Pythagorean theorem to find this. 

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Let's call this KWP, and so let's see.

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We have KP squared plus the other leg [squared] is 8 [squared] 

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is equal to the hypotenuse squared. Remember the Pythagorean theorem.

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So working and solving for KP, 17 squared minus 8 squared is 225.

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And the square root of that I believe is 15, but let me check. Yes.

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So the missing side of the second triangle, this side, is 15.

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So now we need to find in the smaller triangle we need to find RT and ST.

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That's going to be easy because we know a side in the smaller triangle is 1.

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 1 corresponds to the 8, so the proportionality constant in this example

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would be 8 over 1 which of course is 8.

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So in order to find the sides in the smaller triangle we take the side of the larger triangle, divide by 8.

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So RT corresponds to 15, so I have 15 divided by 8.

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And ST, the hypotenuse here, corresponds to this hypotenuse,

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so I take 17 divided by 8 to find those missing sides in the smaller triangle.

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There are two special right triangles whose properties are important for you to know.

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Let's look at the first one. The first one is pi over 4, pi over 4, pi over 2

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or a 45, 45, 90 right triangle. I've got one drawn here. 

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We can see that the legs are equal. Remember, if the legs are equal we know 

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that this angle is pi over 4 and this angle is pi over 4, 

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so we've got an isosceles right triangle. We know the legs are the same length, 

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so I've labelled them with the letter a.

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Now I've labelled the hypotenuse c. So we can use the Pythagorean theorem,

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and we can find the length of c. So we know that c squared, the hypotenuse squared,

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is leg squared plus leg squared. Well a squared plus a squared is 2 a squared. 

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Now to find c remember we're only choosing the positive. We're going to take the 

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square root of 2 a squared. a and c are both positive. So this is going 

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to be the square root of 2 times a. 

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So in any pi over 4, pi over 4, pi over 2, or 45, 45, 90 triangle, 

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the hypotenuse is always the square root of 2 times the length of the leg.

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That's an important thing for you to know.

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If you need to find the leg you would take the hypotenuse and divide by the square root of 2.

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But this relationship is very important for you to know. 

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We can let a be any number. So let's draw that same triangle, and we'll let a equal 1. 

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So the legs are 1. And then what would the hypotenuse be? 

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Well, it's going to be the length of the leg, which is 1, times the square root of 2

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which is the square root of 2. And of course you can use the Pythagorean theorem, 

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and this would be the square root of 1 squared plus 1 squared which of course 

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is the square root of 2. This is a very important triangle for you to remember. 

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You're going to use it throughout the course. So remember this whole triangle, 

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the pi over 4, the ones on the legs, and the square root of 2 for the hypotenuse.

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Now let's look at another special right triangle.

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Let's look at a second important special right triangle that you should know. 

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It's the pi over 6, pi over 3, pi over 2 or the 30, 60, 90 right triangle.

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To get the sides we're going to start with an equilateral triangle. Remember that 

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means all angles are equal and all the sides are equal. So I've got the tick marks

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denoting all the sides are equal. We're going to start by drawing a height 

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from one of the vertices. Now when it's an equilateral triangle, that height bisects 

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both the angle up here and the side down here. So that pi over 3 angle got bisected 

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into pi over 6 and pi over 6. Of course this is a right angle because it's a height.

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Now if we let this side be 2a, then remember we've bisected this side, so

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this little piece is a. So now we have that pi over 6, pi over 3, pi over 2 triangle.

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Let's just draw that piece, that triangle. Pi over 6, pi over 3,

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this is 2a, this is a. And let's call this h, this is the height.

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So we can find that third side in the right triangle using the Pythagorean theorem.

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We have h squared plus a squared equals hypotenuse squared, 2a squared.

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Well let's see, when you square 2a what do you get? You get 4 a squared.

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Now we'll subtract a squared from both sides. We get 3 a squared. 

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So h is going to be the square root of 3 a squared. Remember a is positive.

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So this is going to be the square root of 3 times a.

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So the side opposite the pi over 3 is going to be the square root of 3

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times the side opposite the pi over 6. So this is an important relationship as well.

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So what you see in this triangle, the hypotenuse is 2 times the leg opposite

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the pi over 6 angle. So the hypotenuse is always 2 times this side.

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And the leg opposite the pi over 3, or 60 degree angle, is always the square root of 3

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times the leg opposite the pi over 6 angle. So again these are important relationships 

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for you to remember in this right triangle. Hypotenuse is always 2 times 

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the leg opposite the pi over 6, and the leg opposite the pi over 3

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is always the square root of 3 times this side.

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Now we can let a be any number. So what we're going to do is, what if we let a equal 1?

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Then what are we going to get in that triangle? Well, here's my pi over 3.

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Here's my pi over 6. So we're letting a be 1. So this side will be 1.

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Hypotenuse will be 2, and this side will be the square root of 3 times 1 which of course 

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is the square root of 3. This is another, or your second, important special right triangle

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that you should remember. It's important that you know you've got to put your angles 

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in your triangle. You have to remember to put the 1 opposite the pi over 6, 

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the square root of 3 opposite the pi over 3, and the 2 on the hypotenuse.

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We're going to use these a lot throughout this course.

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Now let's look at some examples where we use these special right triangles.

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Let's look at some examples where we find the lengths of missing sides using

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techniques we learned earlier about similar triangles. 

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Let's look at this first right triangle. We see we've got a right angle. We see

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one angle is marked as pi over 4, so the third angle in that triangle has to be pi over 4. 

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So this is one of our special triangles, our pi over 4, pi over 4, pi over 2 triangle.

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We want to find the lengths of the missing sides. Well, you should remember 

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one of those triangles, the pi over 4, pi over 4 triangle. We can let the lengths 

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of the legs be 1, and remember the hypotenuse is the square root of 2. 

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So this is that special triangle you should have memorized. And now we can use that

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proportionality constant we talked about earlier. We want to find the length of BC. 

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That's pretty easy because we know the legs are equal, so I know BC has to be 5. 

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Now we want to find the hypotenuse. Well, the proportionality constant is 5 

00:20:11.971 --> 00:20:18.960
because 5 and 1 correspond. So to find AB we're going to multiply 

00:20:18.964 --> 00:20:22.218
the proportionality constant, 5, times the square root of 2, 

00:20:22.220 --> 00:20:29.468
so AB will be 5 square roots of 2. Let's look at another example.

00:20:29.470 --> 00:20:35.218
So here we see a right triangle again. We've got one angle marked as pi over 6, so

00:20:35.220 --> 00:20:39.468
we know the third angle in that right triangle has got to be pi over 3.

00:20:39.470 --> 00:20:44.468
So this is our special triangle, pi over 6, pi over 3, pi over 2 triangle. 

00:20:44.470 --> 00:20:51.860
Well, you should remember your special triangle where this is pi over 6, the side opposite 

00:20:51.862 --> 00:20:57.468
is 1, the hypotenuse is 2, and the side opposite the pi over 3 is the square root of 3.

00:20:57.470 --> 00:21:04.234
So that's a triangle you should have memorized. And now we see that 11 corresponds to 2,

00:21:04.236 --> 00:21:13.000
so here I can find the proportionality constant is 11/2. So in order to find BC 

00:21:13.002 --> 00:21:17.214
we're going to multiply 11/2 times the corresponding side in this triangle. 

00:21:17.216 --> 00:21:23.720
So BC will be 1 times 11/2 which of course is just 11/2.

00:21:23.722 --> 00:21:29.236
Now we'll find the last side, AC. Well AC corresponds to the square root of 3, 

00:21:29.238 --> 00:21:33.219
so we're going to take 11/2 again, the proportionality constant, 

00:21:33.221 --> 00:21:38.468
times the square root of 3. So it's going to be 11 square roots of 3 over 2 

00:21:38.470 --> 00:21:42.740
for the length of AC. Let's look at one more. 

00:21:43.719 --> 00:21:48.968
This last example again we see one angle is a right angle. We see one angle is pi over 3. 

00:21:48.970 --> 00:21:53.231
So we know the third angle in this triangle is pi over 6,

00:21:53.233 --> 00:21:59.468
and we know the leg opposite the pi over 3 is the square root of 15. Again, 

00:21:59.470 --> 00:22:04.725
we can just use this triangle here. So the sides that correspond are the square root of 3

00:22:04.727 --> 00:22:09.216
and the square root of 15 because they're opposite that pi over 3. So in this example,

00:22:09.218 --> 00:22:14.460
the proportionality constant is the square root of 15 divided by the square root of 3

00:22:14.462 --> 00:22:19.469
which is the square root of 5. So in order to find the other missing sides, 

00:22:19.471 --> 00:22:22.469
we're going to multiply the corresponding side in the smaller triangle 

00:22:22.471 --> 00:22:30.738
by the square root of 5. So BC will be 1 times the square root of 5 which of course

00:22:30.740 --> 00:22:39.218
is just the square root of 5, and then AB, the hypotenuse, is going to be

00:22:39.220 --> 00:22:46.960
2 times the square root of 5. So we've found all the missing sides 

00:22:46.962 --> 00:22:50.730
of these special right triangles. This is very important because you will use 

00:22:50.732 --> 00:22:55.467
throughout this course these triangles and find missing sides. 

00:22:55.469 --> 00:22:58.968
So it's important that you remember those relationships, you remember these triangles,

00:22:58.970 --> 00:23:03.222
and you can find missing sides pretty easily.

00:23:03.224 --> 00:23:09.976
You should practice all the examples that we've done today so that you can do them on your own.