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Area of a Triangle Using Trig
Good day students in this clip were going to be going over how to find the area of the
triangle using trigonometry Psalm let's go ahead and take a look at the formulas that
we are going to be using to find the areas and then what applied into the three examples
okay so let's write down the formulas so there are three cases of three groups of cases there were going to be looking at the
first one is the SAS situation you have a side to side the included angle SAS the formula
for the area in this case is simply one half a be signed see okay where the included angle
of your secant see you see or another we can look at this is one half of the side as the
side times the sign of the included angle which is represented by a okay right so SAS
is very easy because when you have an SAS situation all you simply do is plug in the
two sides and included angle into this equation and you're done okay when you have an SSS
situation you cannot use trigonometry for this you have to use a special formula known
as he was formula for this case the area is given by the square roots of the cemetery
S times the semiperimeter minus a time the semi perimeter minus B time the semi. Minus
C okay you might wonder what is the semi perimeter of the semi perimeter is just what you gets
when you added three sides a plus B+ C and divided by of two write so this is the first
case the street for the second case is also pretty straightforward but what of if we have
S S A S is a is equivalents to a SS which of the we want to look at it or if we have
him AAS which is equivalent to S AA what of if we have the situation is right here what
the have these case is of what you have to do is you have to convert these cases in twenty
S A S situation using degrees loft sine circuit is very convenient using law of science right
and we convert these into an SAS situation then you simply use the SAS formula to finish
the problem off okay so these are the cases are going to be looking that's right to lets
take a look at question number one for question one with the fire of this triangle right here
so let's leave all the sides the relevant sites this is A so call this a this is B so
call this b this is C utterly needs c in this problem you see wine in minutes now recall
SAS situation is very easy and direct right SSS requires the use of the of yours formulae
and then the other ones require more work so what's configuration do we have here can
it is and what it is well let's say we have assigned in the have an angle and have a site
this is excellent because this is an SAS situation which requires very little work because the
have everything that we need okay right so let's Psalm right of the formula for the area
here so for number one the area is going to be one half S S sign a to two S's are a and
D sign the included angle is C okay Lenny partitioned my workspace here so we can get
confused this is the region of area for the formula as camera put my solutions to the
left or right okay so now we have the formula written down now is going to apply the formula
to this triangles right so you have a is equal to one half a is of eleven p.m. thirteen sign
one oh six okay and eighty plug this in your calculator with the Iranian is to the nearest
tenth your answer is going to be approximately sixty-eight point seven square kilometers
kilometer square right to that will be your area's right let's take a look at question
number two write so what situation do we have here all is label what the have we have a
side the have an angle in the have the angle okay this is a S AA or AAS situation right
so what are we going to do here well we just require some work as indicated earlier the
we have Palm SS AAS our S AA we have to convert it into an SAS situation so we can apply this
formula okay so let's see him which side you want to find will depend on the angle that
you want to use okay so let's say we want to use SAS lets you want to find the site
right here the site okay so let's label the triangle and then we can add it isolate the
side we want to find and express this using a variable right so this is Y this to be y
right here this is why in this side right here is going to be what's to side is going
to be x okay so this is S AAS now we have to use loft signs here in order to find little
X right so you not to use loft signs will going to be outside in the pairs to pairs
the first pair we must know the measure of the side of the angle in the opposite of angle
the second pair must know either the site or the angle right so in this case you want
to find little X so is going to be using the formula the start and what we looking for
X over sign of big X equals now the next I chose this pair because I'm looking for X
okay the next angular selector must know of both the upper the upper lower case right
so let's see which pair satisfies that the have the wise we know both the angle and the
site so you have little why over sign of big why okay right you notice in this equation
I must have only one unknown doing unknown that's permissible as x, does the X right
here can I determine what the value this angle is answer is absolutely we can use elementary
geometry to determine of this angle because the sum of the angles in a triangle is one
eighty okay so to find the us to find and the X update X is equal to one eighty minus
the sum of the other two fifty-seven plus one oh two okay buddy do the calculations
you final that X is equal to twenty-one degrees so let's do that in their twenty-one degrees
right so what we looking for again one of find little X so we have an SAS situation
and we can find the area so let's umm plug everything in this formula right here
so that I have of with the X over sign of big X which is twenty-one equals Y which is
eight over sign of big why which is fifty-seven to get X by itself will multiply what sites
by sign twenty-one sign twenty-one eighty plug is entire expression in your calculator
you have little X equals three point four one eight four okay so let's do that in our
triangle but OS is three point four one eight right is that our answer the answer is now
we just found one site the problem ask is to look for the area okay so what are we using
here we want to use SAS right S AAS that's will the user follows we use why because he
a little X okay so find the area was partitioned my workspace again so to get confused when
we looking for the area right here that log anymore so the area a is going to be one half
S S find a so S is why the other S is X sine of the included angle which is Z right so
let's our input the values so have one half of why which is eight X which is a three point
four one eight sign Z which is one oh two write to plug all these into your calculator
will have the approximate value of thirteen point four square inches or inches square
right to that's the area for the triangle in number two right us take a look at number
three what's situation do we have here for number three we have a side side and the side
this is S S S okay so what form what do we use here the member in this case a can use
of trigonometry because will have any of the angle so that have to use the he was formula
right so let's you collect that he was from right is again hear someone for the area of
a travel given SSS is can be the semiperimeter S times the square root of on-site cemetery
there goes in cyberspace again so we have of area equals the square root of the semiperimeter
S times S minus in this case of this is P so the call this p this is R McAuliffe r this
is Q so close q so very S minus p times S minus q times S minus r okay so this is of
the area formula so let's figure what the semiperimeter is semiperimeter is of the sum
of the three sites keep is Q plus are divided by two okay to us into the values we can have
twelve plus of these two cures for topless for plus twelve divided by two write to at
this all up you going to have of twelfth is twelve is of twenty-four plus four is twenty-eight
okay twenty a divided by two your fourteen to that is semiperimeter so we ready to plug
everything to the formula so the have a the area of this SSS triangle equals the semiperimeter
which is fourteen times fourteen minus key forty minus twelve times fourteen the semiperimeter
minus Q which is of four times fourteen which is the semiperimeter minus twelve right so
that's that's the expression right and then this is the simplify this will have the square
root of fourteen times for two minus two is to for two minus four is tan fourteen minus
twelve again is to okay and work this out the exact area is going to beef the squared
of five hundred and sixty which is approximately twenty-three point seven what's the next here
meters miles of square right to this is your area using the heroes formula okay so that's
that the thanks so much for taking the time to watch this presentation because the free
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