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7.3 Notes

Added Jan 21, 2015 •

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okay 7.3 geometric means that we're going to use geometric means to solve parts of right triangles this is another way to lower right triangle besides the Pythagorean theorem we've talked about geometric means and I told you we would go a little more in depth of where it came from in this is what this first aren't you so the first thing is a geometric sequence there's one in which each consecutive number is found by multiplying the previous number buying given Factor cake so you've done sequencing before the first numbers 5 in your factories 3 so start with V and then multiply by 3 each time 5 * 3 is 1515 * 3 is 45 Titans 3 is 135 and you can keep going on and make a sequence your factors 3 now consecutive numbers of the sequence 4 proportions in fuel tank about the geometric means that we learned if we put 5/15 that is going to equal 15 5 and if you do your cross-multiply you'll see that those are equal ratios again we can start with the 15 instead of the 5:15 45 equals 45 over 135 so we just used the next set of numbers right here 15 stop 35 okay no note 15 over 45 is equal to 45 over 135 if you do your cross multiplication note that the denominator  1 fraction becomes the numerator the next this is known as geometric means and this is where they get this number  okay so it's basically a sequence in your finding the average within the sequence Okay so geometric means between two positive numbers A&B is the positive number we're at a over x equals x over this is what we learned in the last chapter same thing do the same like so funny geometric means between 2 and 10 you put two over x equals x over 10 so you're finding what number will be in the middle of those two if you're doing this sequence so again you cross multiply 2 * 10 is 20 x x x x squared take the square root of both sides next equals now remember 20 does it have a perfect square in it and it does it's four times five so remember can take the square root of the Force 2 and the 5 has to stay underneath so it's to the square root of 5 so in a sequence it would be to square root 5 in in 10 okay now find the geometric means between a 32 and the square root of 2 we're going to have a lot of square root stuff now okay so you would do 8 √ 2 over x equals x over the square root of 2 so when you cross multiply this way now I'm going to ride it out 8 the square root of 2 times the square root of 2 when you're multiplying square roots you multiply what's underneath together and what's on the outsides together so the 8th is the only thing on the outside now the square root of 2 times the square root of 2 is the square root of four cuz two times two is four but what is the square root of 2 the square root of the square root of 4 is 2 so that's 8 x 2 which is so this cross right here actually equals 16 this cross is x squared take the square root of us sides x equals 4 so it would be 8 the square root of 2 for the square root of 2  okay on this page we take geometric means and we set him up in some similar triangles geometric means are two ratios equal so that your proportion remember we can use proportions on similar triangles okay so if we have a similar triangle I don't know if is already labeled or not but this is be the Sissy how much is a 7.5 tells us that if we draw an altitude from the right angle to the hypotenuse play Let's go ahead and draw that show from the right angle straight down to the hypotenuse right there if you can see it Formed to right triangle see there's a right triangle in here's a right triangle and then we have the big right triangle so there's actually three right turn what is when we do that those three right triangles are all similar so the altitude from the right angle to the hype forms two right triangles similar to the original trailer I separated out all three of those triangles over here okay so let's say that we're this hits right here is in so here's my big triangle that we started with there's BC and a okay now look at this small triangle this small triangle will match up with that so let's label it accordingly the right angle is in the shorter side is a and to the longer side is C okay now let's match up this singer triangle right here  okay you're 90 degree angles in that's going to BC and that one's going to be be another a little close together okay now here's the deal these are all going to be similar to each other no in this big triangle that we started with  think about it this is your altitude right now this is your short leg  then we're going to start naming the parts of a right triangle long leg short leg unless it's isosceles is the only time we're not going to have one that is long turn in the other in this is your long leg okay  altitude splits the hypotenuse this is your hypotenuse this whole thing into two parts we're going to name them Segment 1 and segment 2 okay what's a lots of parts to look at stop and maybe pause and look at that again we have a short leg a long leg the hypotenuse is Segment 1 segment 2 and your altitude self over here  you can figure out by AA that these are all going to be similar to each other they all have a 90° angle in common and they all share an angle a is equal to a so these two right here are similar by AA and  likewise over here B same is that one so these two are similar by AA  these two are summer by those two are similar by AA so they're all lower to each other and then these share see which they all have in common so if you'll notice all three triangle what are similar by AA similarity K so therefore we can use this geometric means in these ratios going to be a little hard to see right now but if you'll stop and really think about it it works okay and you can ask me questions on that in class okay so here's the important part okay so Durham 7.6 the geometric means of the altitude so first thing is draw this out inside so now we have a short leg SL a long leg cake in the two segments that it makes let's make what's always do the short leg and Segment 1 and the long leg is connected to segment 2 here's our altitude okay so the altitude  is the geometric means the two segments of the hypotenuse should I just take geometric means of segment 2 in Segment 1 I will get the altitude  okay so if I were to set that up in this triangle let's label it BC and Ann just like the one above it if I do a n which is segment one instead of side by side  the geometric means between Segment 1 and 2bm segment to the geometric means of those is your altitude now your altitude is CN and answer the same so altitude goes there in there so if I set up a problem there's my two segments right there and here's my altitude right there find a geometric means you got your answers so here's an example here's segment one here segment tooth this is your altitude the altitude is the geometric means between the two segments which is 3 and 7 here so altitude is H cross multiply 3 x 7 21 h x h h squared take the square root of those sides h 21 is not have a perfect square so it's just the square root of 21 and that your answer okay here's to more problems using that same formula not remember basically that formulas is that the altitude  is the metric means between Segment 1 and segment 2 so if you use this formula for all of these problems that you know the parts of the hypotenuse altitude this will work  okay so when this problem you have one segment  you have the altitude and you have to find me what are segments so if you plug it into this formula you'll have it now let's just say that this is segment one and we'll make this one segment 2 okay so Segment 1 X over the altitude equals the altitude over segment to that's her ex now just cross multiply and do the algebra 6 x 4 x 4 16 divided by 6X equals to .67 so this right here is 2.67 so if you needed whole hypotenuse you would add the six in the 2.67 and get 8.67 okay so let's go on to this one now I'm this one we have the whole hypotenuse labeled is x in this little section is 25 we need to know what segment 2 is labeled you can put it back into that formula so remember the whole thing is X the that part is 25 so this will be the whole thing at - 25 so that will be our Segment 1 in our segment to okay now using that same form of segment 125 over the altitude equals the altitude over segment 2 now just do your algebra distribute the 25 so that's 25 x -2 5 * 25 is 625 equals 10 * 10 is 100 add the 6:25 over here  so 25 x 725 / 25 x equals 29 soap I know this is not labeled but directly X's 29k so if this part is 2529 - 25 is 4 for this part remembered go with how it's not what it looks like okay so now we have another theorem just like this but it's the fun the legs that one was involving the altitude this one is involving the let them 7.7 geometric means of the leg it says that each leg is the means between these two things it is between the whole hypotenuse the segment of the hypotenuse that is adjacent that leg okay so if I draw my altitude because that's where you get the segments of this whole hypotenuse here's Segment 1 here's segment 2 remember this is our short leg long leg this is the whole hypotenuse soap  if you're doing the short leg you're going the geometric means between the segment adjacent and the whole hypotenuse if you're doing the long leg you did a geometric means between segments to the one beside it and the whole High okay so if we use the labeling that we have on here this is in the short leg is csoc just going to go here in here and it's between the segment that is adjacent an  the whole hypotenuse which is a be so it will be between a be so if I ride that out the long way this will be the short leg the short leg is between segment one in this drawing and the whole hypotenuse okay now the lonely is going to be the same but instead of Segment 1 it's going to be segment 2 because segment 2 is adjacent to that long leg okay so in this the long leg is CB so CB will go right here and right here and it's between the segment that's adjacent to the long leg BM  I mean the whole hypotenuse again be a or a B tank soap this is long leg is the geometric means between segment 2 and the whole hypotenuse now look at the difference between these two short leg segment one long leg segment to hypotenuse is the same for both okay when geometric means for this problem if you notice that they give us we need to find the short leg X and we need to find the long leg the remember X will be between the sided to Jason and the whole hypotenuse so what is the whole hypotenuse will whole hypotenuse you just add those together that's 10 so X is the geometric means between 3 and 10 why is the geometric means between 7 and 10th so let's do the short leg first that will be X short leg so  the short leg X is the geometric means between the going to Jason three in the whole hypotenuse 10 cross multiply that's 30 equals x squared take the square root of Knights x equals  square root of 30  square 8:30 doesn't have a perfect square so it's just Square Reader 30k are long leg  is why it's the geometric mean between decided Jason 7 in the whole hypotenuse 10 so it's between 7 and 10 cross x squared equals 70 take the square root of both sides and 70 doesn't have a perfect square in it either so why is the square root of 70 now you can also add decimals so this is our short leg this is our long leg okay now look at this one turn on this one you have to find the length of the hypotenuse and you know one part of it but you don't know the other part so think about it if I already know this leg and the one attached to it that would fit into one of my formulas and that seems to be the short leg so remember the shore plague is the geometric means between the segment that's adjacent and the whole hypotenuse okay so we're going to use the short leg one short leg is the geometric means so it goes here in here now that is between the segment adjacent 16 and the whole hypotenuse m all you have to do now is cross multiply 16 m equals 2520 is 400 by 16  but actually that is 25 so m is 25 so therefore actually find the other part from their know what I'm going to tell you about this lesson is if you will go back and memorize those two formulas the geometric means of the leg in the geometric means of the altitude you'll have no problem with this but if you don't memorize those two formulas this is going to be a little bit tougher okay now your assignment for this page is going to be page 453 number 13 through 24 30 and 33 that remember you have class time on say to work on these and I'll answer questions and help you through it  I hope this went well

Uploaded by Pam Chastain.

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