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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