hello class welcome to the geometry lesson 7.4 similarity and right triangles so our essential question today is how in a right triangle what is the relationship between the altitude to the hypotenuse triangle similarity and the geometric mean and our goal is to be able to use similarity and the geometric mean to solve problems involving right triangles so our vocab today is going to be the geometric mean so we're going to talk about right triangles in this lesson and a theorem known as the altitude uh two hypotenuse theorem and so what it involves is uh you have a right triangle here um right triangle abc and we're going to draw an altitude coming straight um from the right angle in vertex c down to the hypotenuse a b uh so when you draw an altitude to the hypotenuse of a right triangle it turns out that you form three similar right triangles and this is what we've just done so we drew an altitude to the hypotenuse and you notice that there are two three right triangles um that can be formed from this there's the one on the left the one on the right and then the original large right triangle before it was divided turns out that these three triangles are similar uh and if we look at them in more detail because we have two angles that are congruent between each pair of angles we can claim that they're all similar by the angle angle similarity theorem all right so theorem seven four is the altitude to the hypotenuse of a right triangle divides the triangle into two uh triangles that are similar to the original and to each other and so um basically what we were stating earlier is that uh if you're going from the right angle to the hypotenuse with an altitude then you form three similar right triangles so let's uh go ahead and investigate this in example one so if we want to find the missing angle so for for part a we'll call this x and we'll call this y and what we'll do is we'll use the fact that these are similar triangles to be able to solve this so we're going to use the triangle on the left hand side and we know that well we we know according to this that pqr which is the bit this big triangle is congruent to qsr so in fact we're going to use the triangle on the right according to the similarity statement however we could use the triangle on the left as well but given that they give us the similarity statement we can go ahead and use it so um it states here that if we want to find x well that's going to be q s on the smaller uh right triangle and this uh similarity statement on the right hand side stands for the smaller right triangle so uh qs um according to the similarity statement it corresponds to pq and uh this makes sense because this is the longer leg and this is the longer leg on the big right triangle so that makes sense so x is to 20 as now we're going to look at the another side in the right triangle so if we look at for instance the 15 side we can see that this fifth uh well let's look at the nine side actually because uh let's see uh 15 here uh for this example if we look at the nine side for instance this nine side which is sr relates to qr in the big right triangle um so that relates to this side so uh that's the perfect set of sides to pick because we have two unknown values and we can use this proportion to solve for x so we're going to cross multiply here leaving us with 15 x equals 9 times 20 or 180 and divide by 15 on both sides so go ahead and divide that 15 goes into 18 once and we'll have uh looks like it's about 12 so x is equal to 12. and we'll go ahead and replace this with 12 and use this to solve for y so let's go ahead and use y here now it says here that if we're trying to solve for ps and if we use this similarity statement there's multiple ways that you can solve this but uh this time i'm going to um use well i'm going to use this entire side because that involves y plus nine but you could also use the um small the triangle on the left side to solve it however i'll use y plus nine and we know that y plus nine uh in this big right triangle corresponds to uh 15. on this triangle here so y plus 9 is pr which is this one and so that corresponds to qr all right so so it corresponds to 15 as the um if we were to use this side 20 this side 20 corresponds to uh this side which is nine and you can look at this from the similarity statement it says here that pq relate it corresponds with uh qs so it should actually be this one right here so 20 is to 12 rather rather than nine and now we can go ahead and cross multiply here so we have here we're going to distribute this 12 into y plus 9. we'll do that in the next step and that's equal to 15 times 20 which is 300 now we'll go ahead and uh distribute here so we get 12y plus 9 times 12 is we have 108 which is equal to 300 we'll subtract 108 on both sides so that's 192 and then finally we'll divide by 12 on both sides so we get y equals we'll do this on on the calculator here real quick we've got 16. so y is equal to 16. all right so looking at the next example we want to relate the altitude to what we call the geometric mean so for this particular triangle uh we know that we have three similar right triangles because of the uh theorem uh seven four and in this case we want to find cd so um let's call from we know from 7 4 that the triangles are similar so what we can do is set up a proportion where ad is to cd in this case 6.4 is to this unknown which we can call x or we'll just call it cd and cd is proportional to beating if you were to look at the cd from this side that's proportional to db on the right hand side all right and if we cross multiply we can see that we have cd squared equals a d times bd and the to get rid of the square root to get rid of the squared we take the square root on both sides uh and so we can see that cd is the square root of the product of a d and bd if we were to plug in the numbers we would get a cd a value of 4.8 now it turns out that's this cd which is the square root of this product is known as the geometric mean so the geometric mean of two numbers uh a and b is given by the following formula you take the product of you take the square root of the product of the two numbers this is different than what we known as know as the geometric or the arithmetic mean which is normally what we think of as the average so for example if we wanted to take the average of 10 and 20 also known as the arithmetic mean we simply add the two numbers and divide by how many numbers we have which in this case is two to find the geometric mean we simply take the square root of the product and notice that these numbers are pretty similar since they are a certain type of average so it makes sense that they'll be similar to each other so uh this geometric mean is important as it relates to the theorem seven four and this is corollary one a result of this theorem is that um because we know that the triangles are similar kind of like how i showed you in the other slide we know that we can think of solving for c d the altitude as taking the geometric mean of in this case a d and b d and those are the lengths of these segments of the hypotenuse so in other words in order to just go straight to the answer we can simply take the square root of these two components and get the answer right away for this altitude a d c d the other result of this corollary of this theorem is corollary two which states that if um i want to uh find the legs of the original right triangle so if i wanted to find ac that's simply going to be the geometric mean of ad and the total hypotenuse if i wanted to get bc i simply take the geometric mean of db and the hypotenuse the total hypotenuse and so these are the formulas we can just use the geometric mean rather than having to set up the proportions all right so let's go ahead and show you how this works given triangle rst we want to find rt and st so we know that the triangle rut is similar to rts according to the theorem so we can set up a proportion um so we know that r u is proportional to rt and rt and this side is proportional to our s so we can go ahead and set up the numbers here uh rt which is um you know our unknown here is equal to um r u or rt squared is r u times rs uh and rs you recall is this whole thing so we can plug in the numbers uh we can take the square root and then uh plug in the numbers and we see that we get 9.6 plus 5.4 which is a total of 15. that's this total hypotenuse 15. and so we take the product of this number and 15 as i mentioned earlier in the corollary two and then we get the geometric mean of rt and this is the geometric mean uh from that theorem we can also use uh the concept of the geometric mean for the next part as t so in order to do st we can think of it as um which is this part we can think of it as the geometric mean of this part and the total hypotenuse of 15 and we'll show you this with proportions of course so we know that su correspond is proportional to st and st is proportional to sr so we can set up the numbers here and then find that we could take the square root of su times sr as we mentioned earlier and we plug in the numbers and we get nine all right so as we mentioned those are their geometric means so uh we're going to find the following values using uh the idea of geometric means so if i wanted to get uh x for instance uh well we know that for example this is an altitude and so if i wanted to get this altitude i know that the altitude is equal to the geometric mean of the components of the hypotenuse that are broken up into so we know that 6 is going to be the geometric mean of those two components there then we can use that to solve for x so we're going to square both sides here we get 36 equals and the way to get rid of the square root is with the squared so they cancel and so we're going to distribute this x here and that leaves us with x squared plus 5x and so we got a quadratic equation here we're going to subtract 36 just to make one side equal to zero and turns out we can set up a chart with um negative 36 and five the two numbers here and that's going to be nine and four nine and negative four nine times negative four is 36 negative 36 and then you add those to get five so we can factor this into zero equals x plus nine x minus four using the two numbers here and uh set each of these equal to zero so x would have to be negative nine to make this zero and x would have to be four to make that zero so we have two solutions but obviously only one of them can be viable and it's going to be x equals four because we can't have a negative number for a side so therefore x is equal to 4. all right so in order to find y we're going to make use of the geometric mean of the leg here so to get y well y which is the leg is the geometric mean of the this part next to it the closest thing next to it which is x and the total hypotenuse now the total hypotenuse is you've got to add these together so it's going to be x plus x plus 5 which is 2x plus 5. so it's going to be x times 2x plus 5 here and we already know what x is so we can plug in x into this equation so we have the square root of four times eight plus five so we have the square root of four times thirteen and the square root of 4 times 13 is roughly 7.2 now let's go ahead and find z now z is the geometric mean of the closest part next to it which is x plus five and the hypotenuse so it's going to be the product of x plus five and two x plus five and then we plug in the value for x so 4 plus 5 is 9 and then 8 plus 5 is 13. we get a value roughly 10.8 to the nearest tenth all right guys that's gonna do it for the video i hope you found this useful i hope you uh took a few couple of things from this lesson as usual i'll see you in the next one