Pythagorean Theorem: A Step-by-Step Guide | Find a Missing Side Length Using the Pythagorean Theorem

welcome to math with Mr Jay [Music] in this video I'm going to cover how to find an unknown side length of a right triangle using the Pythagorean theorem we will go through how to find the length of the hypotenuse this is where we will start and then we will move on to finding the length of a leg now remember we use a squared plus b squared equals c squared with C always being the hypotenuse and then A and B being the legs if we know the lengths of two sides of a right triangle we can use the Pythagorean theorem to find the length of the third side let's jump into our examples starting with number one where we have a right triangle and we are given the lengths of the legs the hypotenuse is unknown remember the hypotenuse is the longest side of a right triangle and the side across across from or opposite of the right angle so let's use a squared Plus B squared equals c squared in order to find the length of the hypotenuse now again C is always the hypotenuse so this is C and then the legs are A and B let's make this a and this B now keep in mind it does not matter which leg is a and which leg is B now we can plug in the side lengths that we are given and work through the equation in order to find the length of the hypotenuse so a is eight meters so let's plug in 8 for a so 8 squared plus b is 6 meters so plug in 6 for B so 6 squared equals c squared 8 squared gives us 64. plus 6 squared gives us 36. 64 plus 36 is 100 so 100 equals c squared our next step is to isolate C in order to figure out what C equals so again we are just working through this equation to figure out C since C is being squared let's take the square root in order to isolate C whatever we do to one side of the equation we must do to the other so we need the square root of 100 as well c is now isolated so we have C equals and then the square root of 100 is 10. so C equals 10. let's rewrite that with C coming first so C equals 10 and this is meters so the hypotenuse measures 10 meters and that's our missing side length let's move on to number two where we have a right triangle and again we need to find the length of the hypotenuse we are given the two legs we have 10 feet and seven feet so let's use a squared plus b squared equals c squared in order to find the hypotenuse let's call this leg a this leg B and then the hypotenuse is C now we can plug in and work through this equation so for a we have 10 feet plug in 10 for a so 10 squared plus 7 feet for B so plug in 7 for B 7 squared equals c squared 10 squared that's 100 plus 7 squared that's 49. equals c squared 100 plus 49 that gives us 149 equals c squared now we have 149 equals c squared so our next step as we work through this equation is to isolate c c is being squared so let's take the square root in order to isolate C whatever we do to one side of the equation we must due to the other so let's take the square root of 149 as well we end up with an isolated C on the right side of the equation and then as far as the square root of 149 149 is not a perfect square so we do not end up with a whole number answer C equals the square root of 149. now we can write the square root of 149 as an approximate rounded decimal the square root of 149 is irrational the decimal never stops and it doesn't repeat so again let's round it and we will round it to the hundredths place so the square root of 149 gives us 12. point two zero six and again that just continues on so I'm going to write it to the thousandths place and then round it to the hundredths so let's round it to the hundredths place we have a 6 to the right in the thousandths so this rounds up so the square root of 149 is approximately 12 and 21 hundredths so C is approximately 12 and 21 hundredths and this is feet so the hypotenuse is approximately 12 and 21 hundredths again this is feet so there's how to find the length of the hypotenuse let's move on to finding the length of a leg here are a couple of right triangles and we need to find the length of a missing leg let's jump into our examples starting with number one where we have a right triangle with one of the legs given and then the hypotenuse given so one of the legs measures five yards and then the hypotenuse measures 13 yards we need to find the length of the missing leg we're going to use a squared plus b squared equals c squared in order to find the length of that missing leg now as far as that right triangle let's call this leg a and this leg B remember it doesn't matter which leg is a and which is B and then the hypotenuse is always C now we can plug in what we are given and work through this equation in order to find the length of the missing leg now a is the missing leg so let's keep this as a squared plus b squared well B is 5 yards so let's plug in 5 so 5 squared equals well C is 13 yards so let's plug in 13 and this is squared so that gives us a squared plus 5 squared gives us 25 equals 13 squared gives us 169. so let's continue on working through this equation and we are working to isolate a we are working to figure out what a equals let's subtract 25 from this side and whatever we do to one side of the equation we must do to the other so subtract 25 from the right side of the equation as well the 25s on the left side of the equation cancel each other out so we have a squared equals and then on the right side of the equation 169 minus 25 gives us 144. so now we have a squared equals 144 and we need to continue to look to isolate a and figure out what a equals since a is being squared we can take the square root in order to isolate a whatever we do to one side of the equation we must do to the other so we need to take the square root of 144 as well a is now isolated so a equals the square root of 144 is 12. so a equals 12 and this is yards so the missing leg measures 12 yards let's move on to number two where we have another right triangle and we need to find the length of the missing leg we are given the hypotenuse which is 9 meters and then one of the legs which is four meters so let's use a squared plus b squared equals c squared let's call this leg a this leg B and then again the hypotenuse is always C now we need to plug in what we are given and work through the equation in order to find the length of the missing leg so let's start with a squared so we need a a is four meters so let's plug in 4. so 4 squared plus b squared well B is the missing leg so let's keep this as B squared equals c squared C is nine meters so plug in 9 so 9 squared let's continue to work through this equation 4 squared is 16. plus b squared equals well 9 squared is 81. now we can subtract 16 from the left side of the equation and whatever we do to one side of the equation we must do to the other so let's subtract 16 from the right side as well the 16s on the left side of the equation cancel each other out so we have B squared equals and then 81 minus 16 gives us 65. since B is being squared we can take the square root in order to isolate B whatever we do to one side we must do to the other so let's take the square root of 65 as well B is now isolated so b equals and then as far as the square root of 65 65 is not a perfect square so we do not end up with a whole number answer b equals the square root of 65. now we can write the square root of 65 as an approximate rounded decimal the square root of 65 is irrational the decimal never stops and it doesn't repeat so again let's round it and we will round it to the hundredths place so the square root of 65 gives us 8.06 to and then that decimal continues so I'm going to write it to the thousandths place that way we can round it to the hundredths place there is a two in the thousandth so this rounds to or is approximately eight and six hundredths B is approximately eight and six hundredths or 8.06 and this is meters so the length of the missing leg is approximately eight and six hundredths and again this is meters so there you have it there's how to find an unknown side length of a right triangle using the Pythagorean theorem I hope that helped thanks so much for watching until next time peace