in this video we're going to talk about how to calculate the missing side length of a triangle and we're going to 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 the 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 it's the hypotenuse it's across the 90 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 six and we can make x equal to b so let's replace a with six b with x and c with 10. so now at this point all we need to do is solve for the missing variable 6 squared is 6 times 6 which is 36. 10 squared is 10 times 10 which is 100. now to get x squared by itself we need to subtract both sides by 36 100 minus 36 is 64. now to calculate the value of x we need to take the square root of both sides the square root of x squared is x the square root of 64 is eight so this is the missing side of the triangle it's eight units long now let's try another example kind of similar but a little different so we're still going to 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 replacing a with 9 b with x and c with 16 this is what we now have 9 squared that's nine times nine that's eighty-one sixteen times sixteen that's two hundred and fifty-six subtracting both sides by eighty-one we have 256 minus eighty-one which is 175 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 need to find a perfect square that goes into 175 and 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 though is 25 because we can rewrite 175 as 25 times 7. seven quarters is 1.75 now the square root of 25 is 5. so this is the final exam i mean the final answer in its 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 given two sides of the right triangle we're given one side and an angle and 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 sohcahtoa 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 let's write it as opposite divided by hypotenuse so sine is basically a ratio of two sides of a right triangle the cup 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 h is 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 50. 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 use the calculator and make sure it's in degree mode not in radian mode sine of 30 is what's my calculator is in radian mode i've got to put in degree mode sine 30 is one half now let's cross multiply so we have 2 times x which is two x and that's equal to one times fifty now to get x by itself we need to divide both sides by two so x is fifty divided by two which is twenty five so that is the missing side left 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 100 divided by the adjacent side which is x so now you need to use a calculator to calc 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 1.732 something something but it helps to know that tangent is root 3 which i'm going to write as root 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 100 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 three so we're gonna have a hundred 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 and 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 going to 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 use lower case 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 going to 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 what 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 11.9175 or we can just say approximately 11.9 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 going to have a triangle that is not a right triangle but this time we're going to have an angle we're going to 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 angle c 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 ten so this is going to be nine squared plus ten squared minus two times nine times ten 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 divided by 2 or 0.8660254 so now let's do the algebra so 81 plus 100 is 181 and 180 divided by 2 is 90. so we have 90 square root 3. and let's type this in 181 minus 90 square root 3. that is 25.11542732 so now let's take the square root of both sides to get the value of x so x is equal to 5.01 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 nine this is 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 0 this portion becomes 0. 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 the missing side lengths 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 missing 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 total part tangent is opposite over adjacent tangent or opposite to 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 y that's our first equation now tangent thirty if you plug that in that's equal to the square root of three over three so let's cross multiply so we have y times root three is equal to three x 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 going to 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 100 plus y now let's type in tan twenty twenty tangent twenty is point three six three nine seven and let's cross multiply so this is 1 times x which is x and that's equal to 0.36397 times 100. 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 100 and to y so 100 times 0.36397 you just need to move the decimal 2 units to the right and you'll get 36.397 and then plus 0.36397 times y now i'm going to move this number over here the square root of 3 divided by 3 is .57735 and then times y moving this to the other side it will switch from positive to negative and that's equal to 36.397 so 0.57735 minus 0.36397 gives us 0.21338y now we need to divide both sides by 0.21338 to get y by itself so y is going to be 36.3 divided by 0.21338 and so y is 170.57 so now that we know what y is we can plug in here to get x so if you multiply the y by root 3 divided by 3 you'll get that x is 98.48 so that's how you can calculate the missing 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 in 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 eight so eighteen is nine times two and then we can break it up into the square root of nine and eight times two is sixteen the square root of nine is three the square root of sixteen 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 3 24 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 divisible 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 going to use this right triangle let me use a different color and notice that z is the hypotenuse of that right triangle so we're going to have a squared plus b squared equals c squared let's make a equal to eight 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 divisible by 16. if you divide 2 or 8 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 to 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 eight 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 z in order to solve this equation or 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 to 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 eighteen and 8 or 26 so we can say that y squared plus z squared is equal to 26 squared let's call this equation one equation two and equation three so what i'm going to do is i'm going to subtract equation one by equation two so i'm going to rewrite equation one and then i'm going to multiply equation two by negative one so this would be a negative eight 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 three so i'm going to write it below equation three y 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 going to 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 936. dividing both sides by 2 we can see that y squared is half of 936 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 root 13. so we have the value of y now we can calculate the value of x and 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. and 18 squared is 324. so 468 minus 324 is 144. 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 squared 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 using algebra to solve for the three missing 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