Special right triangles
30-60-90 Triangles II More examples using 30-60-90 triangles.
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- Let's continue with the 30, 60, 90 triangles.
- So just review what we just learned, or hopefully learned--
- at minimum what we just saw, --is if we have a 30, 60, 90 --
- and once again, remember: this is only applies to 30, 60, 90
- triangles --and if I were to say the hypotenuse is of length
- h, we learned that the side opposite the 30-degree angle,
- and this is the shortest side of the triangle, is going to be
- h over 2, or 1/2 times the hypotenuse.
- And we also learned that the longer side, or the side
- opposite the 60-degree side, is equal to the square
- root of 3 over 2 times h.
- So let's do a problem where we use this information.
- Let's say I had this triangle right here.
- It's a 90-degree triangle; let's say that this
- is 30 degrees.
- And we could also figure out obviously if that's 30, this
- is 90, that this is also 60 degrees.
- And let's say that the hypotenuse is 12.
- The length is 12 and we know that this is the hypotenuse
- because it's opposite the right angle.
- What is the side right here?
- Well, is the side opposite the 60-degree angle, or is it
- opposite the 30-degree angle?
- It's the 30-degree angle that opens into it, right?
- I drew this triangle a little bit different on purpose.
- The 30-degree angle opens up into this side, and it's
- also the shortest side.
- We learned that the side opposite the 30-degree angle is
- half the hypotenuse, so the hypotenuse is 12;
- this would be 6.
- And this side, which is opposite the 60-degree side, is
- equal to the square root of 3 over 2 times the hypotenuse.
- So it's the square root of 3 over 2 times 12, or it's just
- equal to 6 square roots of 3.
- Another interesting thing is, of course, the longer
- non-hypotenuse side is square root of 3 times longer
- than the short side.
- I don't confuse you too much.
- Let's do another one.
- Let's say this is 30 degrees-- it's our right triangle --and I
- were to tell you that this side right here is 5, what is
- the length of this side?
- Well first of all let's figure out what we have.
- 5 is which side?
- So if this is 30 degrees, we know that this is
- going to be 60 degrees.
- So 5 is opposite the 60-degree side, and x is the hypotenuse.
- Since x is opposite the 90-degree side, it's also
- the longest side of the right triangle.
- So we know from our formula that 5 is equal to the square
- root of 3 over 2 times the hypotenuse, which in
- this example is x.
- And now we just solve for x.
- We can multiply both sides by the inverse
- of this coefficient.
- So if you just multiply 2 times the square root of 3-- can
- ignore this --we get 10 over the square root of three here.
- And, of course, this 2 cancels out with this 2.
- This square root of 3 cancels out this square root
- of 3 is equal to x.
- And now if you watched the last couple of presentations, you
- realize that this could be the right answer, but we have a
- square root of 3 in the denominator, which people don't
- like because it's an irrational number in the denominator.
- And I guess we could have a debate as to
- why that might be bad.
- So let's rationalize this denominator.
- We say x is equal to 10 over the square to 3; to rationalize
- this denominator we can multiply the numerator and the
- denominator by the square root of 3.
- Because as long as we multiply the numerator and the
- denominator by the same thing, it's like multiplying by 1.
- So this is equal to 10 square roots of 3 over square root of
- 3 times square of 3; well that's just 3.
- So x equals 10 square roots of 3 over 3.
- That's the hypotenuse.
- I know I confused you.
- And, of course, if this is 10 square root of 3 over 3--
- that's the hypotenuse --we know that the 30-degree side-- this
- is 30 degrees --we know the 30-degree side is half of
- that, so it's 5 square root of 3 over 3.
- Anyway, I think that might give you a sense of the
- 30, 60, 90 triangles.
- I think you might be ready now to try some of the level two
- Pythagorean Theorem problems.
- Have fun.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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