0:00:00.600,0:00:02.530
In the last video, we[br]showed that the ratios

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of the sides of a[br]30-60-90 triangle

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are-- if we assume[br]the longest side is

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x, if the hypotenuse is x.

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Then the shortest side is[br]x/2 and the side in between,

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the side that's opposite[br]the 60 degree side,

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is square root of 3x/2.

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Or another way to think about it[br]is if the shortest side is 1--

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Now I'll do the shortest side,[br]then the medium size, then

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the longest side.

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So if the side opposite[br]the 30 degree side is 1,

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then the side opposite[br]the 60 degree side

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is square root of 3 times that.

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So it's going to be[br]square root of 3.

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And then the hypotenuse[br]is going to be twice that.

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In the last video,[br]we started with x

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and we said that the[br]30 degree side is x/2.

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But if the 30 degree[br]side is 1, then this

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is going to be twice that.

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So it's going to be 2.

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This right here is the side[br]opposite the 30 degree side,

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opposite the 60 degree side,[br]and then the hypotenuse opposite

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the 90 degree side.

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And so, in general, if[br]you see any triangle that

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has those ratios, you say hey,[br]that's a 30-60-90 triangle.

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Or if you see a[br]triangle that you

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know is a 30-60-90 triangle,[br]you could say, hey,

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I know how to figure out[br]one of the sides based

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on this ratio right over here.

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Just an example, if[br]you see a triangle that

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looks like this, where the[br]sides are 2, 2 square root of 3,

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and 4.

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Once again, the ratio of[br]2 to 2 square root of 3

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is 1 to square root of 3.

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The ratio of 2 to 4 is[br]the same thing as 1 to 2.

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This right here must[br]be a 30-60-90 triangle.

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What I want to introduce[br]you to in this video

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is another important[br]type of triangle

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that shows up a lot in geometry[br]and a lot in trigonometry.

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And this is a 45-45-90 triangle.

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Or another way to[br]think about is if I

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have a right triangle[br]that is also isosceles.

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You obviously can't have a right[br]triangle that is equilateral,

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because an equilateral triangle[br]has all of their angles

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have to be 60 degrees.

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But you can have[br]a right angle, you

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can have a right triangle,[br]that is isosceles.

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And isosceles--[br]let me write this--

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this is a right[br]isosceles triangle.

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And if it's isosceles,[br]that means two of the sides

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are equal.

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So these are the two[br]sides that are equal.

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And then if the two[br]sides are equal,

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we have proved to ourselves[br]that the base angles are equal.

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And if we called the measure[br]of these base angles x,

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then we know that x plus x plus[br]90 have to be equal to 180.

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Or if we subtract[br]90 from both sides,

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you get x plus x is equal[br]to 90 or 2x is equal to 90.

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Or if you divide[br]both sides by 2,

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you get x is equal[br]to 45 degrees.

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So a right isosceles[br]triangle can also be called--

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and this is the more[br]typical name for it--

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it can also be called[br]a 45-45-90 triangle.

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And what I want to do[br]this video is come up

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with the ratios for the[br]sides of a 45-45-90 triangle,

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just like we did for[br]a 30-60-90 triangle.

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And this one's actually[br]more straightforward.

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Because in a 45-45-90 triangle,[br]if we call one of the legs x,

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the other leg is[br]also going to be x.

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And then we can use[br]the Pythagorean Theorem

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to figure out the length[br]of the hypotenuse.

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So the length of the[br]hypotenuse, let's call that c.

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So we get x squared[br]plus x squared.

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That's the square of[br]length of both of the legs.

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So when we sum those[br]up, that's going

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to have to be[br]equal to c squared.

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This is just straight out[br]of the Pythagorean theorem.

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So we get 2x squared[br]is equal to c squared.

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We can take the principal[br]root of both sides of that.

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I wanted to just[br]change it to yellow.

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Last, take the principal[br]root of both sides of that.

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The left-hand side you[br]get, principal root of 2

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is just square[br]root of 2, and then

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the principal root of x[br]squared is just going to be x.

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So you're going to have x[br]times the square root of 2

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is equal to c.

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So if you have a right isosceles[br]triangle, whatever the two

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legs are, they're going[br]to have the same length.

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That's why it's isosceles.

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The hypotenuse is going to be[br]square root of 2 times that.

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So c is equal to x times[br]the square root of 2.

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So for example, if you have a[br]triangle that looks like this.

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Let me draw it a[br]slightly different way.

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It's good to have to orient[br]ourselves in different ways

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every time.

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So if we see a triangle[br]that's 90 degrees,

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45 and 45 like this,[br]and you really just

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have to know two of[br]these angles to know

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what the other one[br]is going to be,

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and if I tell you that[br]this side right over here

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is 3-- I actually don't[br]even have to tell you

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that this other[br]side's going to be 3.

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This is an isosceles[br]triangle, so those two legs

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are going to be the same.

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And you won't even have to[br]apply the Pythagorean theorem

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if you know this--[br]and this is a good one

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to know-- that the hypotenuse[br]here, the side opposite the 90

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degree side, is just going[br]to be square root of 2

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times the length of[br]either of the legs.

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So it's going to be 3[br]times the square root of 2.

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So the ratio of the[br]size of the hypotenuse

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in a 45-45-90 triangle or[br]a right isosceles triangle,

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the ratio of the sides are[br]one of the legs can be 1.

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Then the other leg is going[br]to have the same measure,

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the same length, and then[br]the hypotenuse is going

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to be square root of 2[br]times either of those.

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1 to 1, 2 square root of 2.

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So this is 45-45-90.

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That's the ratios.

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And just as a review,[br]if you have a 30-60-90,

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the ratios were 1 to[br]square root of 3 to 2.

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And now we'll apply this[br]in a bunch of problems.