More trig examples
2003 AIME II Problem 11 A little trigonometry to figure out the area of a triangle
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- and a right angle at C. So let's just try to draw that.
- let's draw that, let's draw the right angle maybe we have to use some coordinates, so let's draw
- the right angle at the origin.
- So, let's say that this is C right over here.
- AC = 7, so A will be put over here, this distance here
- is going to be seven
- and then we have the hypotenuse of our triangle.
- and this can be
- B, this distance over here they tell us, BC
- is equal to twenty four.
- All right.
- Now point M is the midpoint of AB so point M is right over here let me do that in a different
- color
- point M
- is the midpoint
- it's the midpoint of AB so this distance is equal to that distance
- and D is on the same side of AB.
- So, D is on the same side of AB as C, so C is on this side you can call that
- left of the bottom left-hand side of AB
- So that AD is equal to BD is equal to fifteen.
- So D is going to be some place over here.
- It's going to be
- equal distant between enviada see all of a point equal distant between a and b_ are
- going to sit on a line that looks
- something like that this is the midpoint of abt so do you going to sit right over here
- and it's fifteen away from both they and b
- sold look something like that that distance over their is fifteen
- and then at this distance this distance over here it is also going to be fifteen
- given that the area of triangle cd and
- so the area of triangle cd and this is d so see
- the
- and is the triangle that
- they wanna
- think about given to the area title cbm maybe expressed as and times of squirted andover
- p were and and and p_r_ positive integers
- and an end
- p_r_ relatively prior to that just means you can simplify and and is not the divisible
- by the square of any prime so kind and his
- uh... you've simplify the radical as much as possible
- find and plus and plus peace researcher unified
- the area the area of this green triangle of the cd m over here and so if you look at what
- we can do
- to figure it out so we could try some of the court it's for some of these points
- so displayed right over here a
- is going to be it's excise going to be seven i could rub quarter to cc what i'm doing that
- could be the x-axis
- and in this
- side is on the y_ axis
- quartet for a would be seven
- common zero
- cornett for c_ would be zero com zero and quartet for be would be twenty four
- so i would be zero twenty four
- zero twenty-four so the court date four and would just be the average of d_n_a_
- so the court it for and the average of zero and seven eighths seven house
- and the court that four
- the white cord at governor twenty four and zero is
- toward alpha
- that's fair enough now to what we can figure out about the side so you know this is a right
- triangles or got reaction is always to use the protagonist and we know the side and that
- side
- before one of the red in b
- we could just say that
- we just know that twenty-four squared
- plus seven squared
- is equal to
- is equaled two a v is equal to a be squared
- and twenty-four squared is five hundred and seventy-six plus forty-nine is equal to
- a the squared
- and a c five seventy six post forty nine
- if it was plus fifty would give us two five
- twenty who is a six twenty six points one lesson that's a six twenty five
- so six twenty five is equal to
- a_b_
- squid aired so it be is going to equal twenty five
- so a v
- is equal to twenty five so this distance thus the distance of this big harp on this year's
- twenty five
- half of the distance this is going to be twenty-five over too
- from the dam
- from m_t_a_ is also going to be
- twenty-five over to you
- now the other thing we know
- the other thing that we know
- is that
- am right over here
- amma right over here and the triangle see and is a nice also leads triangle how do we
- know that
- wool and if you look at specs quartet it's excoriate is directly in between the export
- its for c_n_n_ seven have since the average
- this is seven
- this is the url it's important as writer of this directly above the midpoint of this base
- over here
- so this is going to be a nice awfully strangle some after cubic foot the triangle oversold
- this length
- this length of the seems useful 'cause this is kind of the base of the area that of the
- truck
- the base of the triangle we care about
- the kind of you this is the base of cd m
- this is also going to be twenty-five over to you
- twenty-five rotors i softly strangled this is going to be the same is that because we're
- symmetric around this
- right over here
- so c we know one side of this triangle
- but if we could figure out decide over here seems pretty straight forward this is going
- to be a right
- triangle right over here because in this line
- from deep dam is going to be perpendicular to a b
- all of the points that are equal distance
- between in b
- are going to be on the line that is perpendicular
- those perpendicular
- to a_b_
- so this is going to be a right wrangle so we can figure out d_m_
- using the protagonist theorem again
- we get twenty-five over to squared
- twenty-five over too
- school where'd
- plus d_m_ squared
- plus the and squared
- plus the and squared is going to be equal to fifteen square is what he called a hypothesis
- this triangles twenty equal two hundred
- and twenty five
- and so what do we get we get d_m_ squared
- d_n_
- squared is equal to
- to twenty-five minus
- six twenty five over
- four sauce for the soul force of two twenty five or four or to twenty-five with for the
- nominee of the same thing is nine hundred
- nine hundred over four
- in the last video actually figure
- died mistakenly said that it over for was one twenty five agassi boneheaded mistake
- with the number show up again here so to twenty-five obscene niner
- over for
- amra subtract from that
- six twenty five over four
- minus six twenty five
- over four
- and uh... this is he cool too
- cian taryn numerator
- let's see we have not heard my six twenty five would be three hundred minus twenty is
- going to two hundred
- two hundred and seventy-five
- over four
- and so d_m_
- is going to be equal to the square root of
- this citicorp to the square root
- of two seventy five or four to seventy five is twenty five
- times eleven twenty five times twelve would be three hundred twenty-five times eleven
- or for
- so this is going to be equal to five square roots of eleven
- over too
- sodium is five square roots of eleven or her too
- now if we could just figure out
- so when you do here is figure out the height of this triangle right over here we could
- figure out the height of that triangle where we're pretty much done one-half the base times
- a height
- but we don't know who see we could figure that if we know the loft co-signed into something
- over there for another sign of the single right over here
- if we know the sign of that angle the side of that angles going to be this
- over
- the side we just figured out
- sofa to figure out the side of that angle
- then we'd be done or robi very close to being done that's not any obvious way
- but one thing we can do if you look at this bigger triangle overhear salami highlighted
- so if you look at triangle be and c
- redraw triangle bmc
- devote artist
- from destroying right over here and want to get to overload of the triangle b
- amma
- see right over here
- we know this side twenty-five over to we know this side over here twenty five or two
- we know this side over years twenty four
- and we know this
- so what we want to do is figure out the sign of the single right here the sign of data
- of anglesey and d
- now that's hard to do but what we can't do is use a lot of course i had the stayed up
- last ninety
- so let's let me let me redraw are triangle
- so triangle pcm
- i could redraw like this
- i could read right actually nice awfully struggle
- so we have b
- c
- and what is this a will right over here
- it's going to be art fair that we care about plus a ninety degrees
- so this angle right here is data
- data plus ninety degrees
- and this overhears twenty four this is twenty five over too
- and this is twenty five over to you
- if they're using this week a diesel off ko signs to figure out
- to figure out what
- data actually is here
- so listen to that
- raise a little bit of trig identities but love co side so we get
- the opposite to the angle squared so we get twenty-four squared
- twenty-four squared is equal to twenty five over to squared
- twenty-five over to squared plus twenty five over to squared
- plus twenty five over to squared minus
- two times twenty five over too
- times twenty five over too
- times of the co-signed of this angle
- times of misc rollover
- the cosign
- data
- data plus
- data plus ninety degrees now you're saying hey sal you know
- this has it in terms of cosign the state a plus side agrees huckabee figure out the side
- of
- data that's what we actually care about
- to figure out the area of the strike a lecture to figure out the height
- of this triangle
- and to do that you just have to
- maker of the realization that we know the trader identity
- we know the trigger identity the because side of data
- is equal i want you state archives are overloaded a cosigner acs
- is equal to sign
- of nineteen minus x
- so the co-sign if they don't plus nine degrees
- so the co-sign of data plus ninety degrees
- is going to be equal to the sign of
- preferences ninety minus whatever zern nineteen minus data
- minus ninety
- which is are called to
- the nineties cancel out
- sign of negative data
- and will sign a mega date is equal to
- the negative assigned uh...
- so this over here simplifies too
- this is a negative sign the status of the right the scientist data here
- and then put the negative out here in this becomes a positive so what is a simple fight
- with twenty-four squared
- which is five hundred and seventy-six
- five hundred seventy six is equal to
- let's see we have
- loudest i won't skip any steps here so we have twenty five square plus twenty five squared
- this is to times twenty five
- this is too
- two times twenty-five or two
- school where'd
- squared plus
- to times this is twenty five over to squared again two times twenty five over to squared
- times signed uh...
- now we just have to sulfur sign of paid us so this is going to be equal to
- five seventy six
- is the called too
- to trying to play five over to squared
- factoring that out
- one plus sign us data
- one-plus sign of kato or
- we can just divide both sides of the equation by this yeshiva me to simplify it this thing
- overhears
- six twenty five
- four
- for the remarkable i dot by two so this thing over here is the six twenty five
- over too
- source the by both sides of this by six twenty five over to and we get
- five seventy six
- times
- times to over six twenty five
- to smoke applying both sides by the inverse is equal to a new model bible time either
- said this asa cancels out
- is equal to
- one
- plus sinus data
- scientist a that we just subtract one from both sides we get sign of state a
- is equal to five seventy six times to see say six times tues one fifty
- plus a thousand sos one thousand one hundred and fifty two
- over
- six hundred and twenty-five that's that part of their minus one
- recive one mostly six twenty five or six twenty five so minus six
- twenty five
- and uh... this is he pulled to after dual math on the side
- so one thousand one hundred fifty two
- minus six hundred and twenty five
- you get a twelve there's becomes a for twelve months five seven four minus two is to eleven
- minus six is five
- so this is equal to five hundred and twenty seven over six twenty five
- now you might not realize it more in the home stretch alice let's draw
- let me broaden this cd and triangle on that
- is really the focus of the problem of the drug a little bit differently
- let me try it
- you dropped slightly differently so now we know it's a very interesting things about
- cd into the sissy
- this is d
- and this is an we know this side over here
- mr dot is five hafte
- squared eleven
- five over to times the square root eleven that's the length
- of the and
- we also know
- that cm right over here is twenty five over too
- we also know that sign of data here
- we also know that sign of data is equal to five hundred and twenty seven
- over at six twenty five that's what we just figured out
- and we can use that to figure out the height
- the heights of this triangle
- because we know that side is is the opposite over the hype on his if we draw our right
- triangle right here so scientist data
- which is five twenty seven or six twenty five is equal to the opposite is equal to the heart
- of this triangle
- is equal to the higher this triangle over the hype on his over
- five over to square roots of eleven
- so d'amato by both sides of this by five squirts of
- five over to squirts eleven
- and we get
- the high of the triangle is equal to five hundred and twenty seven
- over six hundred and twenty five
- times
- five
- over too
- square roots
- of eleven relatively can divide
- six twenty five divide by five is one hundred and twenty five so you get up
- one here and you have a one hundred and twenty-five over here
- so this is equal to
- five hundred and twenty seven
- square roots eleven
- over
- one twenty-five times too which is two hundred
- two hundred and fifty that's the height now with the area of the triangle it's one-half
- base times height
- area is able to one-half base
- which is twenty five over too
- twenty-five over two times a height
- which we just figured out is
- five twenty seven two square roots of eleven over two hundred and fifty
- the seat twenty five divide the numerator by twenty five by the nominal by twenty five
- years tender
- so this is equal to
- five hundred and twenty seven
- square roots of eleven
- over two times two times ten which is sporty so that's our area the whole problem they
- didn't want to find the area they want us to find and plus and plus p
- and plus and plus pizza they want us to find essentially five twenty seven plus eleven
- plus forty
- five twenty seven
- plus eleven
- plus forty
- so five twenty seven plus the eleven is five penetrated this five hundred thirty-eight
- and then plus forty is five hundred and seventy-eight and were done
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