Intro to 30-60-90 Triangles A few more 45-45-90 examples and an introduction to 30-60-90 triangles.
Intro to 30-60-90 Triangles
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- Sorry for starting the presentation with a cough.
- I think I still have a little bit of a bug going around.
- But now I want to continue with the 45-45-90 triangles.
- So in the last presentation we learned that either side of a
- 45-45-90 triangle that isn't the hypotenuse is equal to the
- square route of 2 over 2 times the hypotenuse.
- Let's do a couple of more problems.
- So if I were to tell you that the hypotenuse of this
- triangle-- once again, this only works for
- 45-45-90 triangles.
- And if I just draw one 45 you know the other angle's
- got to be 45 as well.
- If I told you that the hypotenuse here is,
- let me say, 10.
- We know this is a hypotenuse because it's opposite
- the right angle.
- And then I would ask you what this side is, x.
- Well we know that x is equal to the square root of 2 over
- 2 times the hypotenuse.
- So that's square root of 2 over 2 times 10.
- Or, x is equal to 5 square roots of 2.
- 10 divided by 2.
- So x is equal to 5 square roots of 2.
- And we know that this side and this side are equal.
- I guess we know this is an isosceles triangle because
- these two angles are the same.
- So we also that this side is 5 over 2.
- And if you're not sure, try it out.
- Let's try the Pythagorean theorem.
- We know from the Pythagorean theorem that 5 root 2 squared,
- plus 5 root 2 squared is equal to the hypotenuse squared,
- where the hypotenuse is 10.
- Is equal to 100.
- Or this is just 25 times 2.
- So that's 50.
- But this is 100 up here.
- Is equal to 100.
- And we know, of course, that this is true.
- So it worked.
- We proved it using the Pythagorean theorem, and
- that's actually how we came up with this formula
- in the first place.
- Maybe you want to go back to one of those presentations
- if you forget how we came up with this.
- I'm actually now going to introduce another
- type of triangle.
- And I'm going to do it the same way, by just posing a problem
- to you and then using the Pythagorean theorem
- to figure it out.
- This is another type of triangle called a
- 30-60-90 triangle.
- And if I don't have time for this I will do
- another presentation.
- Let's say I have a right triangle.
- That's not a pretty one, but we use what we have.
- That's a right angle.
- And if I were to tell you that this is a 30 degree angle.
- Well we know that the angles in a triangle
- have to add up to 180.
- So if this is 30, this is 90, and let's say that this is x.
- x plus 30 plus 90 is equal to 180, because the angles in
- a triangle add up to 180.
- We know that x is equal to 60.
- So this angle is 60.
- And this is why it's called a 30-60-90 triangle-- because
- that's the names of the three angles in the triangle.
- And if I were to tell you that the hypotenuse is-- instead of
- calling it c, like we always do, let's call it h-- and I
- want to figure out the other sides, how do we do that?
- Well we can do that using pretty much the
- Pythagorean theorem.
- And here I'm going to do a little trick.
- Let's draw another copy of this triangle, but flip it over
- draw it the other side.
- And this is the same triangle, it's just facing the
- other direction.
- If this is 90 degrees we know that these two
- angles are supplementary.
- You might want to review the angles module if you forget
- that two angles that share kind of this common line would
- add up to 180 degrees.
- So this is 90, this will also be 90.
- And you can eyeball it.
- It makes sense.
- And since we flip it, this triangle is the exact
- same triangle as this.
- It's just flipped over the other side.
- We also know that this angle is 30 degrees.
- And we also know that this angle is 60 degrees.
- Well if this angle is 30 degrees and this angle is 30
- degrees, we also know that this larger angle-- goes all the way
- from here to here-- is 60 degrees.
- Well if this angle is 60 degrees, this top angle is 60
- degrees, and this angle on the right is 60 degrees, then we
- know from the theorem that we learned when we did 45-45-90
- triangles that if these two angles are the same then the
- sides that they don't share have to be the same as well.
- So what are the sides they don't share?
- Well, it's this side and this side.
- So if this side is h then this side is h.
- But this angle is also 60 degrees.
- So if we look at this 60 degrees and this 60 degrees, we
- know that the sides that they don't share are also equal.
- Well they share this side, so the sides that they don't share
- are this side and this side.
- So this side is h, we also know that this side is h.
- So it turns out that if you have 60 degrees, 60 degrees,
- and 60 degrees that all the sides have the same lengths, or
- it's an equilateral triangle.
- And that's something to keep in mind.
- And that makes sense too, because an equilateral triangle
- is symmetric no matter how you look at it.
- So it makes sense that all of the angles would be the same
- and all of the sides would have the same length.
- But, hm.
- When we originally did this problem we only used half of
- this equilateral triangle.
- So we know this whole side right here is of length h.
- But if that whole side is of length h, well then this side
- right here, just the base of our original triangle-- and I'm
- trying to be messy on purpose.
- We tried another color.
- This is going to be half of that side.
- Because that's h over 2, and this is also h over 2.
- Right over here.
- So if we go back to our original triangle, and we said
- that this is 30 degrees and that this is the hypotenuse,
- because it's opposite the right angle, we know that the side
- opposite the 30 degree side is 1/2 of the hypotenuse.
- And just a reminder, how did we do that?
- Well we doubled the triangle.
- Turned it into an equilateral triangle.
- Figured out this whole side has to be the same
- as the hypotenuse.
- And this is 1/2 of that whole side.
- So it's 1/2 of the hypotenuse.
- So let's remember that.
- The side opposite the 30 degree side is 1/2 of the hypotenuse.
- Let me redraw that on another page, because I think
- this is getting messy.
- So going back to what I had originally.
- This is a right angle.
- This is the hypotenuse-- this side right here.
- If this is 30 degrees, we just derived that the side opposite
- the 30 degrees-- it's like what the angle is opening into--
- that this is equal to 1/2 the hypotenuse.
- If this is equal to 1/2 the hypotenuse then what
- is this side equal to?
- Well, here we can use the Pythagorean theorem again.
- We know that this side squared plus this side squared-- let's
- call this side A-- is equal to h squared.
- So we have 1/2 h squared plus A squared is equal to h squared.
- This is equal to h squared over 4 plus A squared,
- is equal to h squared.
- Well, we subtract h squared from both sides.
- We get A squared is equal to h squared minus h squared over 4.
- So this equals h squared times 1 minus 1/4.
- This is equal to 3/4 h squared.
- And once going that's equal to A squared.
- I'm running out of space, so I'm going to go all
- the way over here.
- So take the square root of both sides, and we get A is equal
- to-- the square root of 3/4 is the same thing as the
- square root of 3 over 2.
- And then the square root of h squared is just h.
- And this A-- remember, this isn't an area.
- This is what decides the length of the side.
- I probably shouldn't have used A.
- But this is equal to the square root of 3 over 2, times h.
- So there.
- We've derived what all the sides relative to the
- hypotenuse are of a 30-60-90 triangle.
- So if this is a 60 degree side.
- So if we know the hypotenuse and we know this is a 30-60-90
- triangle, we know the side opposite the 30 degree side
- is 1/2 the hypotenuse.
- And we know the side opposite the 60 degree side is the
- square root of 3 over 2, times the hypotenuse.
- In the next module I'll show you how using this information,
- which you may or may not want to memorize-- it's probably
- good to memorize and practice with, because it'll make you
- very fast on standardized tests-- how we can use this
- information to solve the sides of a 30-60-90 triangle
- very quickly.
- See you in the next presentation.
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