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in this video we're gonna talk about how
to calculate the missing side length of
a triangle and we're gonna go through
many different examples starting from
the easy ones to the much harder ones so
let's start with this example what we
have is a right triangle and we have two
sides of the right triangle to find the
missing side we could use something
called a Pythagorean theorem which
states that a squared plus B squared is
equal to C squared now which of these
numbers represent a which one is B which
one is C C is the longest of the three
sides is the hypotenuse it's across the
ninety degree angle so C is 10 now for
the other two it doesn't really matter
what you make them so we could set a to
6 and we can make X equal to B so let's
replace a with 6 B with X and C with 10
so now at this point all we need to do
is to solve for the missing variable 6
squared is 6 times 6 which is 36 10
squared is 10 times 10 which is a
hundred now to get x squared by itself
we'll need to subtract both sides by 36
100 minus 36 is 64 now to calculate the
value of x we'll need to take the square
root of both sides the square root of x
squared is X the square root of 64 is 8
so this is the missing side of the
triangle it's 8 units long now let's try
another example kind of similar but a
little different so we're still gonna
have a right triangle and the hypotenuse
will be 16 units long and one of the
legs of the triangle is 9 units long
what is the other length of the triangle
feel free to pause the video if you want
to try this so let's say that a is 9 B
is equal to X and C the hypotenuse
that's going to be 16 so let's begin by
writing the formula a squared plus B
squared is equal to C squared
so replace an A with 9b with X and C
with 16 this is what we now have 9
squared that's 9 times 9 that's 81 16
times 16 that's 256 subtracting both
sides by 81 we have 256 minus 81 which
is 1 is 75 and just like before we're
going to take the square root of both
sides now the issue is 175 is not a
perfect square so what do we do at this
point well we need to simplify this
radical we needs to find a perfect
square that goes into 175 the 1 is a
perfect square 4 is a perfect square 9
is a perfect square 16 is a perfect
square the reason for that is 1 squared
is 1 2 squared is 4 3 squared is 9 4
squared is 16 the perfect square that we
need those 25 because we can rewrite 175
as 25 times 7 7 quarters is a dollar 75
now the square root of 25 is 5 so this
is the final exam I mean the final
answer and it's exact form so X is equal
to 5 square root 7 now let's try a
different example so we're still going
to have a right triangle but this time
instead of a given two sides of the
right triangle we're given one side and
an angle that we need to find one of the
legs of the triangle in this case what
do you think we need to do here well we
can't use the Pythagorean theorem
because we need to know at least two
sides of the triangle in this case we
need to use trig there's something
called so Toa for those of you who
haven't seen this expression before the
so part tells you that sine is that is
sine of the angle theta in this case
theta would be 30 sine of the angle is
equal to the side that's opposite to it
X is opposite to 30 but was fried as
opposite / hypotenuse so sine is
basically a ratio of two sides of a
right triangle the cop part in sohcahtoa
is the cosine ratio cosine of the angle
is equal to the adjacent side of the
right triangle divided by the hypotenuse
so a is for adjacent ages for hypotenuse
now for the tangent ratio tangent theta
is equal to the opposite side divided by
the adjacent side now looking at the
triangle that we have which one of these
trig ratios do we need to use sine
cosine or tangent so first we need to
focus on the angle opposite to the angle
is X and we know that the hypotenuse is
15 so we have the opposite side and the
hypotenuse side therefore we need to use
the sine ratio so sine of the angle 30
is equal to the opposite side which is X
divided by the hypotenuse which is 50
now to calculate the value of sine you
could use a calculator you could use the
30-60-90 triangle but to keep things
simple and use the calculator and make
sure it's in degree mode not in Radian
mode sine of 30 is awesome a calculator
is in Radian mode I'm going to put in
degree mode sine 30 is 1/2
now let's cross multiply so we have two
times X which is 2x and that's equal to
one times 50 now to get X by itself we
need to divide both sides by two so X is
50 divided by two which is 25 so that is
the missing side life in this problem so
let's try another similar example in
which we'll have to use a different trig
ratio so keep in mind these work if you
have a right triangle so let's say this
time the angle is 60 this is a hundred
and this is X which trig ratio do we
need to use the only side that we're
missing is the hypotenuse we don't know
the hypotenuse opposite to 60 is 100 so
that's the opposite side the other side
has to be the adjacent side so think of
sohcahtoa Toa
Oh a opposite adjacent so we need to use
tangent tangent of the angle the angle
being 60 degrees is equal to the
opposite side which is a hundred divided
by the adjacent side which is X so now
you need to use a calculator to
calculate tangent 60 tangent 60 is equal
to the square root of 3 if you don't
have a calculator that will give you the
exact answer you're going to get 1 point
7 3 2 something something but it helps
to know that tangent is root 3 which I'm
going to write as for me 3 over 1 now we
need to cross multiply so 1 times 100 is
100 and then we have the square root of
3 times X to get X by itself
divide both sides by the square root of
3 so X is equal to a hundred divided by
the square root of 3 now we have a
radical on the denominator of the
fraction which most teachers don't like
to have it there so we need to
rationalize the denominator we can do
that by multiplying the bottom and the
top by the square root of 3 so we're
gonna have 100 square root 3 and 3 times
3 is 9
the square root of 9 is 3 so this is the
final answer X is equal to 100 square
root 3 over 3 now let's move on to our
next example so this time we no longer
have a right triangle we have two angles
one side that we need to calculate the
missing side how do we do this for a
situation like this you need to use
something called the law of sines first
identify the three angles we're gonna
call this angle a angle B and angle C so
the capital letters are used to identify
the angles the lowercase letters are
used to identify the sides across angle
a is side a across angle B is side B and
across angle C is side C so we'll use
lowercase letters for the sides now
here's the formula for the law of sines
sine of angle a divided by side a is
equal to sine of angle B divided by side
B and that's equal to sine of angle C
divided by side C now we don't have the
angle C nor do we have side C so we
really don't need to use that formula or
that portion of the equation so we're
gonna focus on this part sine a over a
is equal to sine B over B so we know
that angle a is 50 degrees side a is
we're looking for that's X angle B is 40
degrees side B is 10 so let's cross
multiply this is going to be x times
sine of 40 degrees and that's equal to
10 times sine of 50 degrees so next we
need to divide by sine 40
and now we'll get the answer so 10 times
sine 50 divided by sine 40 is equal to
eleven point nine one seven five or we
can just say approximately eleven point
nine so that is the length of X or side
a so that's how you can calculate it
given that situation now let's look at
another example so we're gonna have a
triangle that is not a right triangle
but this time we're gonna have an angle
we're gonna have two sides and the
included angle and we want to find the
side across the angle in this case we
can't use law of sines notice that we
have angle a but we don't know side a
nor do we have any other angles so for
this situation we need to use something
called the law of cosines and here's the
formula C squared is equal to a squared
plus B squared minus two a B cosine of
angle C now for this problem it might be
wise to rearrange the letters
we want this to be Anglesey so that X is
side C and it doesn't matter what the
other two angles are so C is X so
looking for X that means that a in this
example is nine B is 10 so this is gonna
be nine squared plus 10 squared minus
two times nine times 10 and then times
cosine of the angle C that's cosine of
30 so 9 squared is 81 10 squared is 100
and then 9 times 10 is 90 times 2 that's
180 and then cosine 30 cosine 30 is
equal to the square root of 3/2 or 0.866
zero two five four so now let's do the
algebra so 81 plus 100 is 181 and 180
divided by two is 90 so we have 90
square root 3 and let's type this in 181
minus 90 square root of 3 that is 25
point one one five four two seven three
two so now let's take the square root of
both sides to get the value of x so X is
equal to five point zero 1 at least
that's the rounded answer so that's how
you can calculate the missing side of
the triangle if you know two sides and
the included angle you could use the law
of cosines by the way know what happens
if this was 90 degrees if that was 90
degrees we would have a shape that looks
like this
we would have a right triangle and let's
say this is angle C angle a angle B so
then this would be 10 this is still 9
this still B this is still a and then
this is C
so angle C is 90 now looking at this
formula cosine 90 is equal to zero if
you type that in your calculator so
because cosine 90 is equal to zero this
portion becomes zero so that basically
disappears and then we get the
Pythagorean theorem which can be used
when you have a right triangle so you
get C squared is equal to a squared plus
B squared that's why you can only use
this if angle C is 90 or if you have a
right triangle if you don't have a right
triangle then you could use the law of
sines or the law of cosines now consider
this composite triangle that we have
here what would you do in order to find
and the missing side length x and y feel
free to pause the video if you want to
give this problem a shot so what we need
to do is write some equations we have
two myths and variables and we need at
least two equations to solve it so the
first equation that we could use has to
do with the first triangle that we have
here now we could use the tangent ratio
tangent of 30 think of sohcahtoa the Toa
part tangent is opposite over adjacent
tangent the opposite of 30 is X adjacent
to it is Y and this is the hypotenuse
which we don't need to worry about it so
tangent 30 is equal to the opposite side
being x over the adjacent side of Y
that's our first equation now tangent 30
if you plug that in that's equal to the
square root of 3 over 3 so let's cross
multiply so we have Y times root 3 is
equal to 3x and let's get let's get X by
itself so if we divide both sides by 3
we get this X is equal to the square
root of 3 over 3 times y so now we know
X in terms of Y
now what I'm gonna focus on is the
larger part of the triangle so let's
redraw so for the larger part of the
triangle this is X we still have a right
triangle and the base of that triangle
is the sum of these two it's a hundred
plus y and the angle is 20 relative to
the angle 20 this is opposite to it and
this is adjacent to it so once again we
need to use the tangent ratio tangent of
20 is equal to x over a hundred plus y
now let's type in tan 2020 tangent 20 is
0.36 3 9 7 and let's cross multiply so
this is 1 times X which is X and that's
equal to 0.36 3 9 7 times a hundred plus
y
so now at this point we have two
equations and two variables we could
solve by elimination or by substitution
so at this point it's best to solve by
substitution since we have X in terms of
Y we can replace this X with the square
root of 3 over 3 times y and this will
allow us to get one equation in terms of
one variable which means we can now
solve for that variable so let's begin
by distributing this value to a hundred
and 2y so 100 times 0.36 397 you just
need to move the decimal two units to
the right and you'll get thirty six
point three nine seven and then plus
0.36 three nine seven times y now I'm
going to move this number over here the
square root of three divided by three is
0.57735 and then times Y moving this to
the other side it will switch from
positive to negative and that's equal to
thirty six point three nine seven so
0.57735 minus 0.36 397 gives us point
two one three three eight why
now we need to divide both sides by 0.2
1/3 3/8 to get Y by itself
so Y is gonna be thirty six point three
nine seven divided by 0.2 one three
three eight and so y is a hundred
seventy point five seven so now that we
know what Y is we can plug in here to
get X so if you multiply that Y by root
3/3
you'll get that X is ninety eight point
four eight so that's how you can
calculate the missin sides of this
composite triangle now here's another
example what is the value of x for this
particular triangle so notice that all
three triangles that you can draw on
this picture are right triangles we have
the first one where this is X that's 32
that's a triangle
we have another one on the Left where
this is X that's eight and then the
larger triangle with the hypotenuse here
where this is 40 so how can we calculate
X for this particular situation whenever
you have a situation like this where the
altitude goes to the hypotenuse x is
simply the geometric mean of 8 and 32 so
it's the square root of 8 and 32 32 is 8
times 4 8 times 8 is 64 the square root
of 64 is 8 the square root of 4 is 2 and
so the geometric mean of 8 and 32 is 16
now let's think about this why is 16 the
geometric mean of 8 and 32 well if you
multiply 8 by 2 you get 16
and if you multiply 16 by 2 you get 32
that's why 16 is the geometric mean
between 8 and 32 if you have a geometric
sequence 16 will be in the middle of 8
and 32 now let's consider another
example
so let's say that this side is 8 this is
18 and this is X and we also have Z and
Y how can we find x y and z feel free to
pause the video and work on this problem
so we know how to calculate X X is the
geometric mean between 18 and 8 so 18 is
9 times 2 and then we can break it up
into the square root of 9 and 8 times 2
is 16 the square root of 9 is 3 the
square root of 16 is 4 so we know that X
is equal to 12 so once we have the value
of x we can calculate the values of y
and z so notice that Y is the hypotenuse
of this triangle so we could use the
Pythagorean theorem a squared plus B
squared is equal to C squared so a would
be 18 X would be B X is 12 and Y is C 18
squared is 324 12 squared is 144 so
adding those two together that gives us
468 so now we got to take the square
root of both sides now 468 is divisible
by 169
actually I take that back it's the
missile by 36 468 is 36 times 13 and the
square root of 36 is 6 so y is equal to
6 root 13
now let's calculate Z so we're gonna use
this right triangle let me use a
different color and notice that Z is the
hypotenuse of that right triangle so
we're gonna have a squared plus B
squared equals C squared let's make a
equal to 8 B is going to be X which X is
12 and C is Z so 8 squared is 64 12
squared is 144 and 64 plus 144 that's
208 so now we got to take the square
root of both sides 208 is the visible by
16 if you divide 208 by 16 you get 13
and the square root of 16 is 4 so it
helps to find the highest perfect square
that goes into this number and then you
can simplify the radical for those of
you who want more examples on
simplifying radicals just type in
simplifying radicals organic chemistry
tutor and I will explain into more
detail how to do that so this is the
answer for Z now notice that this
problem becomes relatively easy to solve
if you know that X is the geometric mean
of 18 and 8 but let's say you didn't
know that how would you solve this
problem it turns out that just by
knowing the Pythagorean theorem you can
calculate the value of x let's talk
about how to do that so note that we
have three different variables X Y and C
in order to solve this equation or it's
to solve this problem we'll need three
equations to solve for three variables
so let's write those three equations
let's start with this triangle so we
could say that 18 squared plus x squared
is equal to Y squared now moving on to
this triangle we see that Z is the
hypotenuse so we could say that 8
squared plus x squared is equal
z-squared next we could focus on the
larger triangle which is also a right
triangle and this time the hypotenuse is
the sum of 18 and 8 or 26 so we can say
that Y squared plus Z squared is equal
to 26 squared let's call this equation 1
equation 2 and equation 3 so what I'm
going to do is I'm going to subtract
equation 1 by equation 2 so I'm going to
rewrite equation 1 and then I'm going to
multiply equation 2 by negative 1 so
this would be a negative 8 squared minus
x squared equal negative Z squared next
I'm going to add these two equations
notice that x squared will cancel so I
get 18 squared minus a squared is equal
to Y squared minus C squared so now I
can combine this new equation with
equation 3 so I'm going to write it
below the equation 3y squared minus Z
squared is equal to 18 squared minus 8
squared so combining those two equations
notice that the Z squared variable
cancels y squared plus y squared is 2y
squared and that's gonna equal we can
add this up 26 squared plus 18 squared
minus 8 squared
now let's plug in the numbers 26 squared
that's 676 18 squared is 324 8 squared
is 64 676 plus 324 minus 64 that's a 936
dividing both sides by 2 we can see that
Y squared is half of 9 36 or 468 a very
familiar answer and then once we take
the square root of both sides we know
that the square root of 468 is 6 through
13 so we have the value of y now we can
calculate the value of x now let's use
equation one to do that so let's move 18
squared to the other side x squared is y
squared minus 18 squared now we know
that Y squared is the square of this
number which is 468 then 18 squared is
324
so 468 minus 324 is a hundred and
forty-four taking the square root of
both sides we can confirm that X is
equal to 12
that's the square root of 144 so now
that we have X we can calculate Z using
this formula so 8 squared is 64 X square
it is 12 squared which is 144 64 plus
144 is 208 and we know that the square
root of 208 is 4 root 13
and so that's how you can get these
answers without knowing that X is the
geometric mean of 18 and 8 it's by
writing three Pythagorean theorem
equations for the three right triangles
and then use an algebra to solve for the
three innocent variables so that's it
for this video now you know many
different ways to calculate the missing
side of a triangle thanks for watching
