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Pre-Algebra - 6.7: Similar Figures

Added Apr 17, 2021 •

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Finding Measures of Similar Figures by 1) Using Proportions to match Corresponding Parts 2) Setting it up as an Equation y = kx y is what you want k is the scale factor x is what you have if scale factor, SF is <1 you have a reduction if scale factor, SF is >1 you have an enlargement

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pretty much you already used scales we were introduced the we're going to use more scales scale is something bigger or smaller so we'll do a quick review on that scale scale scale Factor let's keep it at scale factor affect around your scale if greater than one you have an enlargement if it's less than 1 you have a reduction it's never negative someone ask is it not know it's a fraction it's some type of a fraction it's either an improper fraction or a fraction fraction so you got to think if it's you know if you have it like a 100 the one that means it's 100 over one that means it's a hundred times bigger than what it starts with whereas if you have a 1 to 4 means it's a quarter of the size 25% smaller so that right there when you change a ratio your scale into a fraction it'll tell you is it an enlargement or reduction is it a is the model bigger or smaller you know people always ask why would you have a bigger model why would you make something bigger that's for science usually they make models of plants or insects that use the one from the amps the whole thing is we can't look at a little ant they make a big giant model of some man you know whatever that bug supposed to be enlargement as a model for you guys to look at to see even a diagram as a model so you have a enlargement or reduction that scale and how do you find measurements so all you so this was the question right here what do you do you take what you so it's always an equation it's what you have meaning the information that you is like what it is you got what you have times the scale factor equals what you want let's say I have the measurement of my aunt and I want to enlarge it by the scale factor cuz that's what I want I want a large it so I'm going to multiply and that's the thing you want you want to read which ones they have which ones they want the word problem will tell you am I going to make it bigger or am I going to make a smaller and what I have right now is it already bigger or smaller or do I want to make it bigger or do I want to make it smaller that's how Math Connects to Lexile what you have and then what you X I said back to this if you're going to make whatever measurement you have bigger you got it and that's the whole thing a lot of people get this right here 101 is it 100 / 1 that's what it should be but what if you want to break it down to back what it was originally that's Lexile right there that means I need to take the reciprocal 1 / 100 I'm not going to write it down cuz that doesn't fit in so you got to say which fraction 2 if I'm going to make it bigger I want bigger that means I have to use fraction that's bigger than one if I want to make it smaller that means I have to take the reciprocal make sure fractions less than 1 what do you have multiplied to the fraction and measurements we usually have in a day always start with like a rectangle or a square start with a rectangle and say is making up some numbers 3 and 105 on the I want to make this I want my scale factor I want it and large scale factor in large let's go I want to make it people choose weird dimensions keep it simple it's going play some weird thing 6 by 5 65 so it's an enlargement right there because it's bigger than one so I want an enlargement so what do I do to find my measurements that's the whole thing you know do that you're that's not a correct thing but you know your new rectangle is going to be bigger so what do you do to each one of these that means you have to change you have to do this to each measurements so this size going to be five times 6/5 and then this one right here is going to be 3 * 6/5 are new measurements right here 5 * 633 / 5 this new side is 6 all of it is bigger just a little bit but it is bigger it's an enlargement this side's going to be funky 3 * 6/5 that's a 18 / 5 or change it to a mixed number what is that 3 3/5 that's bigger but that little bit so our new measurements this is a v v by 5 this at 3 3/5 by 6 yeah so it's about innocence for all of this all you're doing when you're taking new measurements is you're doing this equation for each measurement notice what stays the same the 6/5 it's the constant of proportionality remember how I started with the scale factor is the same as your k let's look at this hey is your constants of proportionality we started early and said that was why / X Y / x equals your constant of proportionality this right here what are you do you multiply every number by Kay to keep everything the same pattern if you have a list of you have a t-chart and a list of exes you're * x x k and there is the equation y equals KX that is the proportional equation all we do is multiply this number to each different X and we get our brand new why so in this case what we had was we had 5 and we had three and each time we met supplied it by that same K 6/5 and we got six and 3 3/5 other different but what's the same constant consistent applied ex by 6/5 if we chose another if we had some other weird shape and we had another measurement we multiply that number by 6/5 and we get another measurement the pattern stays the same constant of proportionality meaning it's the same and that's why K is your scale factor you still factor is a concept that means you multiply you got to keep it multiply everything by 6/5 to get a new measurements so that right there that was the main idea of the last lesson but let's get on with this find the missing measure of similar figures use a scale factor to find the missing measurements of similar figures you scale factor to solve problem a fractal is a metric image that can be divided into parts that are smaller copies of the whole this is all those weird I don't know what you want to call it's a what's a spiral effect and if you notice power keeps on going down all it is just in when you have a mirror in the back of a mirror everything is always people say okay put a beer in back of you and you see like an infinite reflection it goes all the way down and then some people get free all it is is just a replication smaller smaller you know so that's what it is so the concept is corresponding parts of similar figures corresponding matching watching so let me hear this a quick idea of corresponding so let's do a b c d I made a rectangle and then I'm going to make another rectangle smaller one should have been shorter but as to end it right there get rid of that let's say and let's say this is a p w h I went alphabetical on this one I went random on that one corresponding mean matching your point is a goes matches with Point p I see matching cuz I want to use that word right now but let's change it point B corresponds with w point D corresponds with v point do I have to go this way I don't I could go the other way point age corresponds with c and that's what the points so in Geometry we do points but here's another thing sides sides also correspond so we have side PV so segment TV that's the symbol when you talk about a point you just put the letter of the point but when you talk about a side what about PV I'm talking about the side PV that's a segment how do you know if matches cuz I'm making sure they face the same I'll do a triangle on a second show you rectangle would have been a TV if PV if they live in it I'm going to say they are similar this the symbol for similar this rectangle is similar to this rectangle and all the things are turned the same way as hope orientation when you make sure they're facing the same direction will do that with the next example is that with which side good TV is similar or matches with a z and someone just said I don't know if it's a d RDA so there's a thing here is Peter V does it matter which way I start do I go with Peter vrv to feed does the length change good know so that's a geometry concept right there he is the same thing as BP a 2d do I have to go in the same direction when the length when all I'm talking about is the one side no you don't have to match up you should but you don't have to do so I went a d I'm going to go I'm going to go DC first I'm in Lodi to see DC so this side matches with this side but that's the thing noticed I'm going to be moving at back and forth this side do I have to started eating all this side matches with this side do I have to start with me know DC is similar to h v matching size and you got the other one so how do you know they match you they'll tell you what are similar to each other and I'll write down these things but let's use visualization first let's say I give you a triangle and what people like to do the top of the line that's called segment it's a symbol for the word segment a segment in geometry we all say this is a line the correct term is line segment in math a line is actually an infinite concept are there a lot of different things this right here is segment segment that's weird second Imagine Dragons. Sorry and fix that segment and you always used to letters cuz this it says this segment starts at p and ends at V this segment starts at a and ends at d this segment starts at D & N. C the segment starts at h and ends at V you always need two points beginning and an end so segment line over something this right here means line and noticed arrows arrows in math number line a number line goes on forever it's an infinite concept for the segment you can actually cut off and measure I can measure between p&w but there exists a line inside that segment this goes on forever also that's another but that's where people start getting lost in Algebra 2 the concept of infinite answers what you'll get to that don't worry we're not there yet and then there's a third option which is a ray we have a starting point but it goes to Infinity in some Direction that's called a ray this is where you start out this is where you go infant all right so let's use a triangle so you can get your saying so I'm going to say this I got this triangle notice right here I got a short side alongside but I have the long 5 and I'm in let's make it bigger it doesn't have to be a smaller version the next one to be a bigger one and Visually just looking you can literally see this is my short side this is my medium side this is my longest side and that's how you know and here's the thing they might look similar but I've been need to tell you these two triangles are similar in that do you match it up here I would do XYZ and R2 RS and you can literally point from one one to another but sometimes sometimes they will do you dirty and what I mean by that I'm going to do the same thing but instead they turn the triangle round these two triangles are similar but now notice this one's turn around it's not oriented the face with the say play facing the same way there are similar so what goes with what and if you just go left right into our X goes with q but you had to think if you turn it around extra do with our Q is where the short and the medium line meet but X is where the short and the long side Meat so that's why I say thing to do when you're asked to do similar course I'm turning around make sure they're facing the same way and then you can match them up so those are precautions and some people are able set those people visualize the stuff you guys are awesome but when you get to a shape that's like many-sided it's always good to make them face the same way and let's go similar figures are figures that have the same shape but not necessarily play the same size so they have the same shape but not the same size Figure 8 in figure ABCD is similar to figure efgh and that's a similar right there live Zoom see that means similar ABCD is similar to efgh so notice and they made him face the same way but how do you know this because they told you it was similar to be would match with EDF so we go in this or a b c d and literally this also tells you what order to write them out in ABCD go around efgh go around you can't zig zag what do I mean by that I'm going to make a different one if I said how to make a triangle again four-sided figures much better if it goes a b c d your four-sided figure just I'm just making it a b c d correct this is incorrect I'm going to do the same figure try to mess that up who cares a b c d you can't just randomly put them anywhere sorry you can't just randomly put them anywhere when they tell you it's a b c d you mean you here's a here's being here see here and someone said how do you know that's where you start with a I'm just a symbol they'll always give you a little bit more information tell you which point start at all right but notice this is track you go around this is incorrect you can't zigzag ABCD you can't you got to go around when you name this stuff so and figures have corresponding parts meaning matching part matching Parts corresponding needs matching corresponds with EB corresponds with f h corresponds with BC correspond G and then we start looking at the sides visually we can see a d is the shortest side ad corresponds with EDH the longest side is b2c that corresponds with h2g self corresponding parts match up now similar figures here's to ideas some figures corresponding parts what are the parts the parts aren't points there two ideas and jopwell segment parts are angles and segments angles and segments in similar figures angles are with congruent the word congruent is similar to equal equivalent angles corresponding parts angles AR congruent when we go to Geometry talking about shapes angles the shape of the angle is congruent and the symbol for congruent looks like equal but it equal sign with the wave on top that just means they're equal where that's the weird thing about geometry it's super technical very simple but that's where a lot of people get lost the shape of the angles are congruent the measures of the angles are equal when we talk about the measurements equal when we talk about the shape of the angle the angle itself there equal this is a lexile detail geometry is all about the detail that's why so many people get lost they sending me the same thing yes they should they're very similar but that little tweak messes up so if someone wrote equal for the shape it would be more wrong if someone wrote congruent for the measure it would be marked wrong even though both ideas saying these two angles are the same and that's what's messed up no angles are exactly right here noticed that this the angles look exactly the same but look at this the only thing different is that the segment or longer I use a square in a second so what does that mean about the segments the angles are congruent the angles are equal the measures of the angles are equal my fault the side the segments are proportional that means you know it's just either shrunken down version or an enlarged version zoom in zoom out zooming in or zooming out the shape stays the same it just gets bigger and smaller the angles don't change and I'm going to use the eye Square someone type in what's special about a square what's special about a square how do you know what the square I'm assuming there's a square but it's kind of rectangular here I'll be quiet square is a rectangle true all sides are equal I'm going to give you another I'm going to give you another shape that has equal sides but is not a square let's assume those are equal there a diamond a rhombus a rhombus has equal sides but it's a scratch there's one extra element that makes a square a square and it goes in what makes a rectangle a rectangle how do you know some of the rectangle by definition and it has to do with the angles so all you guys are right angles 90 degrees look at this right here this this is a smaller version of that right there's a square all squares are versions of each other so right angles right angles that's why we call it a square because 2 square angle right angle notice this right here zooming in zoning out these angles are perfectly 90° 90° angle stay the same except a zoomed-out version resume zoom in and zooming out it doesn't change the shape what angles give the shape of something big put angles on stuff that's what gives it shape and then we just connect those angles with segments look at this right here from this site using this bottom corner angle this side is this long this angle the same except the side is now extended what did the angle change No it didn't so right there similar figures similar figures have corresponding parts these angles and sides in the same position images with h and Otis day and that's the symbol for angle but that's the thing a lot of people mess up and go angle a less than a no-sew when we noticed that when they printed out the sliced so to give your symbol for angle that some people do this right here they make a little curved like a protractor it makes it look like a protractor off to the side so I'm looking at angle a so that little bit you don't have to but that's an accepted thing you could right angle a is just angle a but it's that looks to you like a less-than symbol put that there to say angle a angle a matches with an e angle C measures angle G matching parts and notice right here sides a dented B matches with eventa s and let's the segment symbol and already gave this right here if two figures are similar then corresponding angles are congruent and noticed right that's the congruent symbol that's how to say that they're matching that means a matches with X we can't use the symbol anymore cuz if we just marked one noticed they put a double one and then a triple one you have four so all right let's go back to where is that other drawing that we had I want to mark it up so let's say this wasn't a rectangle this look like a rectangle but it wasn't a matches with p b matches with w d and no decide you don't have to go in order when you're marking similar with symbols 123 you want to go for here no you can actually stop and do this Car-Mart in there so you could do 4 if you want but you can just stop and just go that and then if there's another angle you put 2 marks that just says which ones match up with which one and we know angles are equal that's saying that this angle matches with this one and not only are they matching their equal that's a symbol of congruence for angles so corresponding angles are congruent or have the same measure and corresponding sides are proportional opposite corresponding angles so what it means is the AC corresponds with XZ but both of them are on the opposite side decide. touching of two corresponding angles I gave this these two sides are right across from two angles that are congruent to each other that last part confuse a lot but you don't need this is important corresponding sides are proportional notice they notice right here and equal equals at school or is congruent to angle B is congruent to why the measure of C is equal the measure of Z but they don't have that that's just good girl but noticed what they do is over XY BC is over YZ proportional set it up as a fraction and when you set up as a fraction that are equal to each other proportional I'm trying to think of something simple to do let's do another example they got theirs alright let's look at our first example right here so the figures are I hear someone saying how do you know they tell you the figures are similar find each missing measure so match up the sides and I'm going to show you properties of portion their lot of different ways of setting it up as long as you use the same logic if we're going from triangle to Triangle this triangles to this triangle you always got to go in the same direction we can zig zag exactly four matches with 12 so I could go for over 12 that is equal to 6 / X Latias have a problem the variable being on the bottom denominator just put it in the corresponding position now that means I went triangle to try and I'm going to show you something else and we're going to get the same thing if we know they're similar I can also do this I can do four matches with I could go to the short side matches with this long side 4 matches with 6 that's totally different but if I use the same logic in the Sun one four matches with 612 matches with x for both of these the cross product is the same 4 * X 4 * that equals 6 * 12 6 * 12 divided by 4 is this sorry 4X equals 6 * 12 + 12 / 412 / 436 * 3x equals 18 which one is 4 and there's other ways of doing the do I have to go do I have to start with this triangle and set it up as a let's go with I could even go bigger to smaller alright I want the Faribault Top matches up with 6 x / 6 I could go if I go bigger to the smaller that means my other I have to match I started with this one to this one this side to this side then I have to match the other ones this side to this side and look at our cross product list look at it now look at these right here x / 6 there's no X over sit on I should have matching up here there's no x / 6 12 / 4 what's a reciprocal version of that one it doesn't matter cross products are the same correspond means not just matching Parts but matching order if you do something in one order you better make sure the next step follows the same pattern order so as a teen so since triangle ABC ABC is similar to Triangle you don't have to write the word triangle you just put triangles in front of 3 you read it as triangle triangle ABC corresponding angles are congruent and they just did a proportion like they decide to go 6 to X over 4 to 12 cross product ends up being the same way try be a look at it looks to look at be alright what kind of this right here look something up some people say resume in 2849 D and 4 but notice always I don't have anything on this side because it's not facing the same way that's why you always want to read what's called a simile a statement rstu matches with WXYZ if you're able to look at it without having to rewrite it great but you should get in the habit of redrawing it facing the same way and how do I do that I will always reach all the first one the first one and this is sketch they don't need to be the same size rstu and they don't need to be different and I'm just going to redraw the same shape who cares if it looks there different it's just a visual aid for me I have the measurements I have things that go with each other r s t u r s t u matches with and then ignore don't even look at this with the statement are you matter with w x y z w x y z in the same way now I know they're facing the same way with the put the measurements back because this are you was 28 that means that matches with wz what's wzr what that's for RS is 49 what's wxro that's now everything's facing the same way I won't get mixed up lie to you lot of you are visual learners you can't do it some of you guys think you could do it in your head but for some reason you start mixing things up it is for Lottie we are all visual Learners and for some of you more so than the other it is very smart to face it turning around reoriented facing the same direction so you can match like I said it's up to you how you want to set up the razors I'm not even look how they set it up I'm just going to go 2249 and I'm so if I went this side to this side that means on this when I got to do the same thing 28 is 249 is equal to 4 to 9:40 notice I went here to hear if I went the side to the top I have to go side to the top some of us at War I want to have the variable on top while that's all right. I'll do another way imma show you get the same cross product I'm going to go d249 notice connecting two different numbers to the variable but it doesn't matter following a pattern D is the 49 that means for would have to be the 28th let's look at both of these cross product 4 * 49 * 4 * 49 that's equal to 28 x d d x 28 28 D notice we get the same thing let's / 7 / 28 My Fault by bike 28 5 / 7 I get 7 / 7 I get 4 / 4 / 4 7 equals and let's go look at it right there 7 feet are how they said they went 49 / D do I have 49 / D on any of mine no but I have the reciprocal on this one and I the proportion when they tell you they're similar it's how you want to follow your logic but as long as you keep on repeating the logic consistently your proportion remember setting fractions equal to each other will still get you the same cross product so that is you're so do 1 a.m. and one be on your own right now I'm going to zoom in to make sure you know what you're doing quickly if you don't want to use the cross product that's fine some people said can I simplify the fraction instead and see if they're equal you can do that do you want to divide and see if they get the same number that works too lot of different ways to do the arithmetic part this to this this to this not as this one's actually facing the same way this one isn't smart thing to do is turn around answer your question in a second I'll go back before we start doing this let me know when you're done with bow so I can move on just put done if you got both done thank you for something it's a weird thing this is like a witches coven everything is it is always responds in threes always the three people maybe I am the head Wizard or Warlock of course I choose the side of evil I always root for the evil I rooted for Darth Vader the dark side status Warframe more fun a lot more fun yes when I check it was Pokemon and I had a choice I would choose Team Rocket yes I would and if when it when I when I bother it Pokemon go I chose the yellow why cuz I know everyone choose either red or blue let me choose the one yeah no that's only according to the cartoon sometimes you're the best just ask look at this right here they didn't have the segment or just have the letters so just like angles just like angles there's one that represents the angle and one that represents the measure so RS segment RS represent the actual line itself actually refers to the actual number price of RS is 49 so that RS refers to the 49 this RS with the white on top refers to the physical line this was like a little stupid little details in Geometry but Kelly that's why the state test when the geometry questions are there you got to look at the little my new details be Sherlock Holmes or or what was it when I was a girl Enola Holmes wasn't that her I didn't watch that Netflix movie it's just whatever her name is all right so let's look at this right here this one is mine because it's facing the same way the orientation is matching that means five goes to third here's the thing I don't want to go 5 / 30 / 5:30 to 5 cuz that's six then that means I would have to go the other way 24 to be 30 / 5 I get 624 / what gets me six B would have to be Wellness at 6 but be 24 / B equals 6 that means right here what should that be that means be equals for sorry 30 / 5/6 and that I'm not using a cross product that's just using Math logic 30 / 5 is fixed that means 24 / some number has equal 624 / 4 Chinese b equals 4 you don't need the cross product but let's just do the cross product just for the heck of it and here's the thing by why not simplify first to make it easier cross product to do listen to the cross product right now 30 b equals 5 * 24 equals 4 / b6d equals 24 Deezer I just said 5 this fraction first man I would rather do this one same thing you don't realize you're still doing proportions right here even though we're simplify first and then it's the same thing right here let's turn this trying around the same way so this triangle looks nice so I got I'm going to draw the first big triangle first play k l j k l and this is a this is 14 and hear what will help is also your congruent angle markers one Mark to Mark 3 Mark 1 mark to mark 123 redraw this trying without putting in the letters and use these angle markers to the white street signs 1 Mark to Mark 3 marks is triangle my one Mark is point G Mi to Mark is H and my three Mark is I and let's look at our measure G 2hi is 10 G to I is 10 g2h is for and h2i is 7 notice this triangle down here has all three sides labeled this only has two sides this is a SAT question or one of those standardized questions ignore this now let's ignore that we don't need that anymore because if you use the number 1010 matches up with nothing there's nothing there they was sometimes a measurement that you don't even use but for some reason you think you need to use it and then you're going to mess up by using it there there to trigger here's a a a a it's on this side that means matches with this at all 10 and then it's this side all right that's the side on this side but notice they're facing different ways when you read Rod a matches with four that means 14 2 + 7 and notice we don't even use 10 in her proportion this is majority of the stuff you're going to use for scales could simplify this that's a 2/1 a equals Ocho and that's a simple as it gets let's go look at them next thing I do have to find it can you come back I do have it though all right let's scale factor the scale factor is the let's read this the scale factor is the constant of proportionality lesson 6 for you learn that portion relations can describe using an equation in the form y equals KX recall the scale factor is the ratio of a length on a scale drawing to the corresponding Mike in the real object that's not always the case to be always the other way around it is also the ratio of corresponding sides in similar figures all right let's go read this an architect is designing a decorative window for the entrance of a new office building using similar triangles triangle ABC is trying similar to Triangle d e f find the length of DF so where is ABC Let's ignore this let's do it a b c and they gave us measurements 1820 5.5 and they didn't give us anything for CD now that's Duke ABC is similar to d e f ignore d f in the picture for now Draw d e f and it doesn't it doesn't need to be perfect a b c d e f now go to the picture and fill in the info they have there the only information we have is that e2f is 9 think about something else similar triangles messed up rap by they don't match up how did they do this and why do they do this and here is another measurement I didn't put that there's 18 cuz I covered that picture part I didn't see there was a teen right there so look at this we're trying to find what did it want find the length of DF so we're trying to find this right here I'm going to call it a text so matcha stop right there let's set it up who cares what they set it up as but you know they went scale factor so let's do it two ways listen to scale factor method and let's do the proportion method so listen to the proportion method on the last problem you are correct we do not use 10 on the way it's just there to screw with you look at all this one equal just you can put a 9 match up the parts so let's do proportion first let's go 9 let's go 18 + 9 is equal to this to this is equal to this this 25 25 2X and notice who 829 that's just a true / 1 2x equals 25.5 thirds my cross product I'm sorry a lot of guys want to say always a oh where you get the equation from like that okay so divided by 2 1/2 of 25 is 12.5 half of 5 is .25 so xdx is 12.75 that's just doing Mental Math right there so then here's the thing I wrote Acts if this was a geometry test I have to mark it wrong cuz it wants the length of DF GF so the length of segment DF is 12.7 not going to trip if you wrote Auntie Quizlet but I'm just saying when you get to a test if you had a multiple choice question and you had both of these this is wrong this was right I'm not going to trip I'm just giving out details right now so let's do the scale factor method what they're saying alright so scale factor look at this right here so they want us to go remember this their skill the ratio what are they say the scale factor where is the scale factor the length of the drunk but they're both in the same thing they decide to go scale factor the given side to the missing side so 18290 the scale Factor equals to okay 18 to 9 and that's what messed up so let me rewrite this me stink 18 to 9 I don't think what cuz I don't like how they set it up no thank you thank you the scale factor is just pretty much what you want times the scale factor no it's what you have have I messed it up 5 times a scale factor equals what you want now look at this what we have is actually bigger than what we want so the matching side is 25.5 what the scale if you multiply by 2 it gets bigger so the scale factor we should use is actually the reciprocal so it should be 1/2 for what we want and what we want is side DS smaller size and that's the tricky part of the scale factor if we set it up one way see what you have versus what we want are we making it bigger or smaller thank you for letting me know and I am about to end it and we won't get like to see Thomas same thing

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