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8.4 Notes

Added Feb 19, 2016 •

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8.4 proportionality theorems now this is just a lesson on more ratio proportion we're going to set up the problem solving the same but here's some terms about when you can use proportions and when you can't so I was talking about a shaped like this last time where you have a triangle inside of triangle you got to triangle you got the big and I stress to you that you needed to do the small side of the triangle over the big side  in the small side convert the big side because then this small side went with this big site now you can just break it up can use these parts instead of the whole side and that's them 8.6 right here triangle portionality theorem or what it says is if a line parallel to one side of a triangle intersex  the other two sides divides the two sides proportionally  now I feel look at this figure right here these are parallel over here these are not necessarily parallel if they're parallel than these parts are proportional you're not parallel then these parts are not proportional so the only time that you can break it up is when it's parallel therefore if Skippy is parallel to CD then it breaks it up into proportional parts we can say that from a to c song from B daddy goes with seeded G over d to g so we have this part put this part in this part goes with this parts of this ratio will be equal to this ratio 8.7 is the converse of that and what it says is that if a line divides two sides of a triangle proportionally then it is parallel to the third side so if this ratio from here to here and here to here equal then that means that YW must be parallel to x z cuz the only time these parts are proportional is when these two lines are parallel so then we would know that these are parallel example one in the die be is parallel to CD so these are parallel ABC is 12 IDI is 9 what is the length of a that's going to be RX so since these are parallel we don't have to do a over 20 we can do 8 with the X12 with the nine you can the switch that up you can do 8 with the 12 x with the knife remember pick a pattern and stick with it so if I do it with the x equals 12 / 9 when you cross multiply you get 12 x equals 72 / 12 x equals 6 now let me show you instead of going play we go this way so if I did 8 over 12 equals x over 9 see my pattern 8 - 12 x - 9 I started with the shorter side first if you look again you still get 12 x equals 72 and X is 6 so either way you set it up it will work there's another example that goes with those two Theron snow this is more about the converse it says in the diagram shown where am in if p q and LM are parallel if they're parallel remembered these ratios have to be equal so let's set up the ratios will do the 18 with the 27 so we're doing this across ratio and the 16 with the 24 are those equal will reduce it 9 will go into 18 - x and none will go into 27 three times for this is 2/3 likewise over here you're going to get change when you reduce by 8 so they have the same ratio therefore yes PQ is parallel to that would make those parallel now it's look at parallel lines right here we have three parallel line and these are going to be two transversals remember transverse was anything that crosses two or more parallel once so here's your transversals cancel transversal what happens to these parts that are cut off it says it's three lines intersect two transversals so what it's saying is that these parts are proportional to each other so A&C would match up being good matchup for a b and c d would match up so you can do a over C equals B over d you can do A over B equals c over D so you can go in this pattern or you can go in this pattern now we have one more serum that we're going to talk about and it's when we have a triangle in a ray that bisects the angle make .9 says if a ray bisects an angle of a triangle so you have that right there then it divides the opposite side into segments whose links are / personal  to the links of the other two side so it divides this side in the same the these two sides the other two sides of the triangle are so in this one if angle 1 is congruent to angle 2 so you have this right advice X then this side over this side is the same ratio is this part of that side to this part of that side so you would have be over ad there's your two other sides equal to BC Over CD now this is very important if I did baby on top I have to do the proportion that is beside a be on top so a B2 ad top to bottom top to bottom can I said this is what your Ratio or your proportional look like let's do some examples with that now in here have kind of a funny looking symbol those are supposed to be angle symbols that got translated weird but it says in the diagram if angle 1 2 3 and 4 all congruent so like here that's alternate interior so that makes these parallel that's all turning in. That makes these parallel there for all three of these lines are going to parallel to each other right here is 32 find is what we're trying to find right there so if you notice that makes all of these parallel because you have alternate interior angles so these portions going to be the same ratio so you can do 21 / x 28 / 32 now you can also go in that pattern like this 21 / 20 Wet 20 x / 32 either way is going to work so if I do 21 / X I go across 28 / 32 cross multiply 28 x equals 21 * 32 is 672 divided by 28 and X is 24 so this right here is 24 the Farmers Lane is divided in by a newly constructed Interstate the distance showed are in meters find the distance between North Border in the South border of the farmer's land so between a and C they want us to find this whole thing will we know this section we don't know this section if these are all parallel then that cuts these off proportionally so this will go with this this will go with this you can find that and add it so if I do $2,000 / 2,500 that'll equal this X over 3000 cross multiply when you cross multiply you're going to get 25000 x equals 6 million and when you divide by that 2500 can you were going to get x equals  2400 so this right here is 2400 add that with your is thousand and your final answer from a to c is going to be 4400 whatever units this is measured in it doesn't tell us that is your final answer  tell us an example for we have a triangle and its been cutting laughing they show that this angle right here has been cut in half it's already Eagle those symbols got left out right here this should be angle and angle to tell you those are equal but it says length of a d so we want to know just X right here so we know that this line Cuts this proportional as the sides so let's make a ratio out of the sides let's go from left to right so we'll do we'll 45 / 54 add this X to this part this part is not life but we know the whole thing so if you need to find a part that you don't know you take the whole thing and you subtract so I'm going to label this one third 3 - x because I would take the whole thing I'm on it now I went from left to right so I have to do X over 33 - x then I'm going to cross multiply to get 54 equals now I have to distribute that 45 here so that is going to give me 1480 - 45 x now solve for x I want to move this 45 over here at at it I'm going to get 99 x equals 1485 / 99 on both sides and your ex is going to be 15 so part is 15 and you can find this part 2 because it's 33 - 15th and you're done with that problem we have one more example that and we have that angle deg is congruent to angle GED GF sorry Instagram angle DG is congruent to gef so here's your angle bisector so it cuts this side into proportional Parts the same as these so to set up your problem you can do 8 over 12 going left to right and then X over we have to do this section again the whole thing is 14 so this is going to be 14 - x cross multiply 12 x 12 equals 8 distributed you were going to get one 12-8 hex add this 8X over here and that's going to give you 20 x equals 1 12/20 so at 5.6 know what I want you to do is to pause  the video I want you to work out these for problems on your then come back and I will be worked out and see if you got them all correctly so for number one you should have gotten x equals 15 because you did 10 / x 4 / 644 / 10 + 6 / X number to you should have gotten not parallel because if I put turn over 16 that's 5/8 enough of the 12 or 18 that's two-thirds and that is not the same so they are not parallel  number three picture of got an x equals 34 because if I put the eggs over the 36 in the 51 over the 54 or you could have done X over 50 36 over 54 for upside down from that number for you should have got an x equals 21 now remembered this segment right here you have to take the whole thing and some this part to get what's left between 8 - 6 so I did the left side or the right side the left over the right cross multiply don't forget to distribute and you get next equals 21 from your assignment that you will have to class time to do and work on his page 454 number 3 through 8 13 through 20 to 25 through 26 and 29 through

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8.4 Notes

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