Similar Triangles

so now we can talk about the main idea for lecture three so what does it mean for two triangles to be the same or what does it mean for two triangles to be similar so we say that two triangles ABC and DF are congruent if all of the corresponding parts are congruent that is the triangles ABC are is congruent to DF if angle a is congruent to angle D angle B is congruent to angle C uh angle e excuse me angle C is congruent to angle F the side AB is congruent to De the side AC is congruent to DF and the side BC is congruent to EF and so if we think about the possible drawings we could get from such a thing we say that two triangles are congruent maybe something like this if when labeled so we have like labels a b c d e f so we require that angle a be congruent to angle D we have that angle B is congruent to angle e we have angle C is congruent to angle F so they have the exact same angles um and also in terms of side lengths we have AB is congruent to De we have that CB is congruent to Fe and we have that uh side AC is congruent to DF that's what it means for two triangles to be congruent to each other all of the corresponding parts are congruent as angles and and line segments and so note that congruent triangles are essentially just the same triangle but possibly in different locations in the plane uh so like the difference between ABC and DF is just translation I moved triangle ABC you could also throw Reflections and rotations in the mix there but the two triangles are essentially the same although they might be oriented in different places in the plane okay a weaker notion than congruence is the idea of similarity we say that triangles ABC and DF are similar if just the angles are congruent to each other so if angle a is congruent to angle D angle B is congruent to angle e and angle C is congruent to angle F we say the triangles are congruent to each other excuse me they're they're similar to each other and we don't use the usual congruent symbol we just draw a little squiggle um to say that two triangles are similar to each other well if two triangles are congruent they definitely are similar but it's not necessarily the case that if two triangles are similar that they have to be congruent for example we could draw a picture like the following so drawing back our labels here so we have AB C and we have D EF and I apologize that this picture is not perfectly drawn to scale but you'll notice that angle a is still congruent to angle B angle e is still congruent to B and angle C is still congruent to F so these triangles are similar to each other but they're not congruent you'll know notice that the side lengths of the two triangles are different the DF triangle is much bigger than ABC but even though the side lengths are bigger they have still the same basic shape because the shape of a triangle is determined by its angles not by the side lengths and so when it comes to similar triangles similar triangles have corresponding parts that are that are proportional so AC and DF are proportional to each other that is the the factor which de is larger to AB is the same as the factor as AC as well and so what we see here is if we look at the proportions here if you take AC the the length there and divide it by DF this is equal to the same thing as AB / de which is the same thing as BC divided EF and these are all equal to some constant so like maybe the triangle is twice as big or three times as big or 1 and a half times as big this is what we get by proportionality these ratios are always the same and so because of this similar triangles using this notion of proportionality is a very very useful tool in trigonometry and calculus and Beyond and so I want to demonstrate some examples of how you can set up proportions for similar triangles so consider the two triangles you see on the screen ABC and DF these triangles are congruent to each other excuse me they're similar to each other because their angles are congr congruent A and D are congruent angles B and E are congruent angles and F and C are congruent angles so these triangles are similar we know the three side lengths of triangle ABC um the side AC is worth is length 16 side AB is 24 and CB is 32 but we only know the proportion for FD well because they're proportional if I take the segment AC over DF this is going to equal the this is going to equal the same fraction for the other corresponding parts so we get that AB / de would equal that and BC over Fe these are all going to equal each other some constant and so let's fill in the information we know so a c is 16 DF is 8 um which if you simplify just that fraction right there you get a two and so that's that constant K this triangle ABC is twice as big as DF and so all the other sides are going to satisfy that same proportion so you get that if we take for example ab ab was equal to 24 and if we take de which we don't know what that is we get that de like so uh 2 is equal to 24 over de that's to say that 2 * de is equal to 24 or D equal 12 so the missing side here was 12 um the other one as well if we just cut this in half that's going to give us the side length for Fe as well so that means the remaining side must be 16 because there's a there's a factor of two uh proportionality between these triangles let me give you some examples that I use in my actual Calculus class uh you can actually see the links to these videos uh on the page right now so these are genuine calculus problems and I just want to demonstrate not the calculus part but why similar triangle become essential to some of these calculus problems so imagine a spotlight is on the ground and it shines on a wall That's 12 M away so here's our Spotlight here's the wall and the distance between them is going to be 12 M so we know about that so there's a man who's who's between the wall and the spotlight and he himself is 2 m tall which I remember from Jurassic Park that's how tall a velociraptor is as well so he's a pretty tall dude not not like unhuman tall but you know he's on the taller side of things so if he's 2 m tall and he's standing between the wall and the spotlight the Sha there's going to be a shadow of the man on the wall so like as you see in this diagram here the the light shines but as it hits the man it obstructs the light and so there's a shadow on the wall here let's call why the height of the Shadow okay if the man is exactly 4 M from the building um how tall is the shadow how tall is the shadow here and so if he's 4 M from the building building that means since the the the spotlight and the wall are 8 m apart he'd be 8 m away and so you can take 12 - 4 that's where this 8 came from just so you're aware so what we want to consider here is consider this triangle right here and I'm going to use a different color to emphasize it we're going to draw this one in red let's look at the triangle that's formed between the spotlight and the man and we're just going to extrapolate it for a second over here if we just look at that triangle then notice that this side right here is eight this side right here is two uh and let me mention that this is a right angle this is a right triangle because the man is standing up right um now let's consider the the triangle that's formed between the spotlight and the wall so this green triangle over here let's bring it over to the side so if we draw that again I'm just going to draw it much bigger the distance between the spotlight and the wall is going to be 12 and then the height over here is y we don't know what it is but what we do know is that this is also a right triangle but it's not just that we know that this angle formed from the spotlight is the angle in both of these triangles so we have a right triangle which have two angles that are congruent then because the angle sum of a triangle always equals 180° if you have two triangles where two of the angles are the same then the third angle must be the same as well in a right triangle the two non-right angles are necessarily complementary to each other so because of that the angle or the triangles have to be similar if a right if two right triangles share a common angle that mean they have all angles the same they're similar so we can set up a proportionality argument here so notice that y over 2 corresponds to 12 over 8 so Y and two go together and 12 and 8 go together so y over 2 corresponds to 12 over 8 so if he times both sides by two to clear the denominators well at least to clear the denominator on the right hand side you're going cuz we're trying to solve for y the height of the the height of the Shadow you get y = 24 8 which 8 goes into 24 3 times and we see that the height of the Shadow would be 3 m let me give you another example coming directly from my own calculus course we have a tank uh of water that itself is a inverted cylind uh circular cone right here so it's kind of like an ice cream cone it points downward like this although this diagram right here we're looking straight onto it and we just see a triangle as this cross-section the height of this cone is going to be 12 M the radius of the cone is going to be 4 M as labeled right here this water there's water that's filled into the tank and so the height of the water is going to be 8 m so the tank is not completely filled the height is going to be 8 m and so particular there's a gap of 2 m at the top of this that we could add two more meters of water into this tank right here so let's figure out what the radius of the water is at this specific height so this unknown value is R what will be the radius CU when you fill when you fill the tank with water the tank as the water as a fluid will take the shape of its tank so what's the radius of the water that has this conle shape right here well again we could make a similar triangle argument so the first one I want you to think about is if we look at the whole tank we could take this whole triangle right here but to make life a little bit easier I'm actually going to take half of the triangle this is an example of an esos triangle and we're going to cut the esles triangle in half which always gives you a right triangle so let's look at this triangle right here pull it off over here so the triangle in yellow uh this side length is going to be the height of that triangle which is 10 m this side length is going to be the radius which is 4 M so we have that it's a right triangle and notice that this angle at the bottom right here is just this angle right here next let's look at just the triangle involving the water again we're just going to take half of it pull this over here to the side we get a triangle that looks like this the height is going to be eight that's given to us the radius we do not know and again this is the same angle in play right here so these angles are congruent right angles are congruent to each other so the other two angles have to be congruent to each other uh these triangles are similar to each other so we can set up proportions so this tells us that R over4 corresponds to 8 over 10 for which we can times both sides by four this would give us that R equals 8 * 4 over 10 which is going to give you 32 over 10 which gives you 3.2 met as the radius of the water at this point in the tank suppose we want to next find the radius R when the height of the water is itself a variable okay what if we allow the gap between the top of the tank and the water level what if we call that X so in the example we did first X if the height was eight then X was equal to two Okay so what if we allow that variable to come into play right here then the height of the water the height of the water is just going to be 10 - x whatever the Gap turns out to be and so that then leads to the proportion that R over 4 corresponds to the height which is 10 - x all over 10 in which case then we can solve for R easy enough we get R is equal to 4 * 10 - x all over 10 four and 10 both have a common factor of two which you can cancel out and so we see see that the radius is equal to 25s * 10 - x which we can then check our answer if we take X to B2 like we saw before uh that would look like 2 fths take 10 - 2 which is going to be 8 2 over 5 * 8 2 * 2 Excuse Me 2 * 8 is 16 over 5 which 16 over 5 is the same thing as 32 over 10 which is the same thing as 3.2 this gives us the answer we did before and so this is the idea you know uh why similar triangles can be very important in a calculus setting calculus cares about like when when you take these quantity and treat it like a variable how does the system change well no matter how you change the system it's always a similar triangle so we can find a formula for the radius depending on the height of the water or in this case the gap of the water and and it's always this form is always set up by a similar triangle argument so for calculus students being able to set up and use a similar triangle argument is very very important which is why we want talk about it in this trigonometry class