45-45-90 Triangle Side Ratios Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2)
45-45-90 Triangle Side Ratios
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- In the last video we showed that the ratios of the sides of a 30-60-90 triangle are, if we assume the
- longest side is X, the hypotenuse is X, then the shortest side is X over 2, and the side in between,
- the side that is opposite the 60 degree side is the square root of three times x over 2.
- Another way to think of it is if the shortest side is one, and I'll do the shortest side then the medium
- side than the longest side. So if the side opposite the 30 degree side is 1 then the side opposite the
- 60 degree side is the square root of three times that. So it's going to be square root of three and then
- the hypotenuse is going to be twice that. In the last video we started with x and we said that the 30
- degree side is X over 2, but if the 30 degree side is one, then it is going to twice that so this is
- going to be 2. This right here is the side opposite the 30 degree side, opposite the 60 degree side,
- and then the hypotenuse, opposite the 90 degree side.
- And so in general, if you see and triangle that has those ratios, you say hey, that's a 30-60-90 triangle.
- Or if you see a triangle that you know is a 30-60-90 triangle you cold say, Hey!, I know how to figure out
- out one of the sides, based on this ratio right over here, and just as an example:
- If you see a triangle that looks like this, where the sides are 2, 2 square root 3, and 4.
- Once again the ratio of 2 to 2 square root of 3 is 1 to square root of 3.
- The ratio of 2:4, is the same thing as 1:2, this right here must be a 30-60-90 triangle.
- What I want to introduce you to in this video is another important type of triangle that shows up alot
- in geometry, and a lot in trigonometry. And this is a 45-45-90 triangle.
- Or another way to think about it is, if I have a right triangle that is also isosceles
- ,so a right triangle that is also isosceles.
- You obviosly can't have a right triangle that is equilateral,
- because an equilateral triangle has all, all of their angles have to be 60 degrees.
- But you can have a right angle, you can have a right triangle that is isosceles.
- An isosceles, let me write this, this is a right isosceles, i-sos-ce-les triangle.
- And, if it's isosceles that means 2 of the sides are equal.
- So, these are the 2 sides that are equal.
- And then if the 2 sides are equal we have proved to ourselves, that the base angles are equal.
- If we call the measure of these base angles X, now we know that X+X+90
- have to be equal to 180. X plus X plus 90, need to be equal to 180.
- Or if we subtract 90 from both sides, you get X plus X is equal to 90.
- Or 2X is equal to 90, OR if you divide both sides by 2, you get X is equal to 45 degrees.
- So a right isosceles triangle can also be called, and this is the more typical name for it:
- It can also be called a 45-45-90 triangle.
- And what i want to do with this video, is to come up with the ratios for the sides
- of a 45-45-90 triangle, just like we did for a 30-60-90 triangle.
- And this one is actually more straight forward.
- Because in a 45-45-90 triangle, if we call each of, if we call one of
- the legs X, the other leg is also going to be X.
- and then we can use the pythagorean theorem to figure out the length
- of the hypotenuse.
- So the length of the hypotenuse, let's call that C.
- So we get, X squared + X squared, that's the square of both of the legs.
- so when we sum those up that's going to have to be equal to C squared.
- This is just straight out of the pythagorean theorem.
- So we get 2X squared is equal to C squared, C squared.
- We can take the principle root of both sides of that.
- I want to change it to yellow, and it's not letting me,, ok.
- Ok, to C squared, now let's take the principle root of both sides of that.
- Principle root of both sides.
- The left hand side, you get, principle root of 2 is just square root of 2.
- And then principle root of X squared is just going to be X.
- So you're going to have X times the square root of 2 is equal to, C.
- So if you have a right isosceles triangle, whatever the two legs are,
- they're gonna have the same length, that's why it's isosceles.
- The hypotenuse is going to be square root of 2 times that.
- So C is equal to, X times the square root of 2.
- So for example, if you have a triangle that looks like this.
- Let me draw it in a slightly different way.
- It's good to have to orient ourselves in different ways everytime.
- So if we see a triangle that is a 90 degree, 45, and 45 like this,
- And you really jst have to know two of these angles to know what the other one
- is going to be.
- And if i told you that this side right over here is 3.
- Actually, I don't even have to tell you that this other side is going to be 3.
- This is an isosceles triangle, so the two legs are going to be the same.
- And you wont even have to apply the pythagorean theorem if you know this,
- And this is a good one to know,
- That the hypotenuse here, the side opposite the 90 degree side,
- is just going to be the square root of 2 times the length of either of the legs.
- So its going to be 3 times the square root of 2.
- So the ratio of the sides and the hypotenuse, in a 45-45-90 triangle,
- or a right isosceles triangle.
- The ratio of the sides are: one of the legs can be 1,
- then the other leg is going to have the same measure, the same length.
- And then the hypotenuse is going to be the square root of 2 times either of those.
- One to one to square root of two.
- So this is 45-45-90, let me write, this is 45-45-90.
- That's the ratios. And just as a review, if you have a 30-60-90
- the ratios were one to square root of three to two.
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