Special right triangles
30-60-90 Triangle Example Problem Using what we know about 30-60-0 triangles to solve what at first seems to be a challenging problem
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- So we have this rectangle right over here,
- we call it, the length of AB is equal to 1
- So that's labeled right over there, AB is equal to 1
- And they tell us that BE and BD trisect angle ABC
- So BE and BD trisect angle ABC
- So trisect means dividing it into 3 equal angles
- So that means that this angel, is equal to this angle,
- is equal to that angle
- What they want us to figure out is,
- what is the perimeter of triangle BED, triangle BED
- So it's kinda this middle triangle and the rectangle right over here
- So at first it seems like a pretty hard problem,
- because you're like, what is the width of this rectangle
- how can I start on this, they've given only one side here
- They've actually given us a lot of information,
- given that we do know that this is a rectangle
- We have four sides and then we have four angles,
- the sides are all parallel to each other
- and the angels are all 90 degrees,
- which is more than enough information
- to know that this is definitely a rectangle
- And so one thing we do know that,
- the opposite sides of a rectangle are the same length
- This side is 1, then this side over there is 1
- The other thing we know is that this angle is trisected
- Now we know what the measure of this angle is
- It was a right angle, it was a 90 degrees angle
- So this divide into 3 equal parts,
- that tells us that this angle is 30 degrees
- this angle right over is 30 degrees
- and this angle right over here is 30 degrees
- And now we see we're dealing with a couple of 30, 60, 90 triangles
- So this is other side over here needs to be 60 degrees
- So that side right over there needs to be 60 degrees
- This triangle over here, you have 30, you have 90,
- so this was one have to be 60 degrees, you have to add up to 180
- 30, 60, 90 triangle
- 30, 60, 90 triangle,
- and you can also figure out the measure of this triangle
- although its not gonna be a right triangle
- But knowing what we know about 30, 60, 90 triangles,
- if we just have one side of them
- we can actually figure out the other sides
- So for example, here, we have the shortest side,
- we have the side opposite of 30 degree side,
- now whatever, if the 30 degree side is 1
- then the 60 degree side is gonna be squared 3 times that
- So this length right over here is gonna be squared to 3
- And that's pretty useful because now we've figured out
- the length of the entire base of this rectangle right over there
- Now we just used our knowledge of 30, 60, 90 triangles
- if you see, if that was a little bit mysterious,
- how I come up with that
- I encourage you to watch that video
- We know that 30, 60, 90 triangles,
- there side are in the range of 1 to the square root of 3 to 2
- So this is 1 this is a 30 degree side
- this is gonna be square root of 3 times that
- and the hypotenuse right over here is going to be 2 times that
- So this length over here is going to be 2 times,
- the side right over here, so 2 times 1 is just 2
- So that's pretty interesting, let's see if we can do something
- similar with this side right over here
- Here the 1 is not the side opposite of the 30 degree side
- Here the 1 is the side opposite of the 60 degree side
- This is the 1 opposite the 60 degree side
- So once again, if we multiply this side by square root of 3,
- we should get this side right over here
- This is the 60, remember this 1, so right over here, this 1,
- this is the 60 degree side,
- so this has to be 1 over square root of 3 of this side
- Let me write this down 1 over the square root of 3
- And the whole reason, why I was able to get this is well
- whatever this side is I multiply it by the square root of 3
- I should get this side right over here,
- I should get the 60 degree side, the beside the 60 degree angle
- or if I take the 60 degree side if I divide it by the square root of 3
- I should get the shortest side, the 30 degree side
- I start with the 60 degree side, divided by square root of 3,
- I get that right over there
- And then the hypotenuse is always gonna be twice
- the length of the side opposite the 30 degree angle
- So this is a side opposite the 30 degree angle,
- the hypotenuse is always twice that
- So this is a side opposite the 30 degree angle,
- the hypotenuse is always twice that
- It is gonna be 2 over the square root of 3
- So we're doing pretty good,
- we have to figure out the perimeter of this inner triangle
- right over here
- We already figured out one length is 2,
- we figure out another length is 2 over square roots of 3
- And then, all we have to really figure out is what ED is
- And we can do that it's because AD
- is gonna be the same thing as BC
- We know that this entire length,
- we're dealing with a rectangle is the square root of 3
- This entire length, the entire length of the square root of 3
- if this part, this AE is 1 over square root of 3,
- then this length right over here, ED is gonna be square root of 3
- minus 1 over the square root of 3
- That length minus that length over there
- And now to find the perimeter is pretty straight forward,
- we just have to add these things up and simplify it
- So it's gonna be two, so let me write this perimeter
- of triangle BED is equal to, this is sort of, perimeter
- I just didn't feel writing the whole word,
- is equal to two over the square root of 3
- plus square root of 3 minus 1 over the square root of 3,
- minus 1 over the square root of 3 plus 2
- And now this just boils down to simplifying radicals
- you can take a calculator out
- and get some type of decimal approximation for it
- Let's see, we have two square root of 3 minus one square root of 3,
- that'll leave us with one, one over the square root of 3
- 2 square of 1/3, I should say, 2 over square root of 3
- minus 1 over square root of 3, 1 over square root of 3,
- and then we have plus the square root of 3 plus 2
- Now let's see I can rationalize this
- if I multiply the numerator
- and the denominator by the square root of 3
- This would get me the square root of 3
- over 3 plus the square root of 3
- which is, I can rewrite that as, plus 3 square root of 3 over 3
- Right I just multiplied this 3 over 3
- Plus 2 and so this give us, this is the drum roll part now
- so 1 square root of 3 plus 3 square root of 3
- and all that over 3 gives us 4 square roots of 3 over 3 plus 2
- Where you can put the 2 first,
- some people like to write the non-irrational part
- before the irrational part
- But we're done we figured out the perimeter
- of this inner triangle BED, right there
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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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