 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 DEF are congruent if all of the corresponding parts are congruent. That is the triangles ABC is congruent to DEF. If angle A is congruent to angle D, angle B is congruent to angle C, 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. 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 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 line segments. And so note that congruent triangles are essentially just the same triangle but possibly in different locations in the plane. So like the difference between ABC and DEF 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 DEF 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 similar to each other. And we don't use the usual congruent symbol, we just draw a little squiggle 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 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 ABC and we have DEF. 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 notice that the side lengths of the two triangles are different. The DEF 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 proportional. So AC and DF are proportional to each other. That is 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 length there and divide it by DF, this is equal to the same thing as AB divided by DE, which is the same thing as BC divided by EF. And these are all equal to some constant. So like maybe the triangle's twice as big or three times as big or one 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 wanna 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 DEF. These triangles are similar to each other because their angles are 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. The side AC 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 gonna equal the same fraction for the other corresponding parts. So we get that AB divided by DE would equal that. And we also get that the last one AC over what's the corresponding side there, FD, these are all gonna equal each other. Oh, we already did AC. Starting again in three, two, one. And BC over FE, these are all gonna equal each other, some constant. And so let's fill in the information we know. So AC is 16, DF is eight. 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 DEF. And so all the other sides are gonna 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. Two is equal to 24 over DE. That's to say that two times DE is equal to 24 or DE equals 12. So the missing side here was 12. The other one as well. If we just cut this in half, that's gonna give us the side length for FE as well. So that means the remaining side must be 16 because there's a factor of two proportionality between these triangles. Let me give you some examples that I use in my actual calculus class. You can actually see the links to these videos on the page right now. So these are genuine calculus problems. And I just wanna demonstrate not the calculus part but why similar triangles 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 meters away. So here's our spotlight, here's the wall and the distance between them is gonna be 12 meters. So we know about that. So there's a man who's between the wall and the spotlight and he himself is two meters tall, which I remember from Jurassic Park, that's how tall a velociraptor is as well. So he's a pretty tall dude, not like unhumanly tall but he's on the taller side of things. So if he's two meters tall and he's standing between the wall and the spotlight, there's gonna be a shadow of the man on the wall. So like as you see in this diagram here, 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 Y the height of the shadow. If the man is exactly four meters from the building, how tall is the shadow? How tall is the shadow here? And so if he's four meters from the building, that means since the spotlight and the wall are eight meters apart, he'd be eight meters away and so you can take 12 minus four, that's where this eight came from, just so you're aware. So what we wanna consider here is consider this triangle right here and I'm gonna use a different color to emphasize it. We're gonna draw this one in red. Let's look at the triangle that's formed between the spotlight and the man and we're just gonna 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 and let me mention that this is a right angle. This is a right triangle because the man is standing upright. Now let's consider 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 gonna draw it much bigger. The distance between the spotlight and the wall is gonna 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 degrees, 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, then they have all angles the same, they're similar. So we can set up a proportionality argument here. So notice that y over two corresponds to 12 over eight. So y and two go together and 12 and eight go together. So y over two corresponds to 12 over eight. So if you times both sides by two to clear the denominators, well, at least to clear the denominator on the right-hand side, you're gonna, cause we're trying to solve for y the height of the shadow, you get y equals 24 over eight which eight goes into 24 three times. And we see that the height of the shadow would be three meters. Let me give you another example coming directly from my own calculus course. We have a tank of water that itself is a inverted 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 gonna be 12 meters. The radius of the cone is gonna be four meters as labeled right here. This water, there's water that's filled into the tank. And so the height of the water is gonna be eight meters. So the tank is not completely filled. The height is gonna be eight meters. And so particularly there's a gap of two meters 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 would be the radius? Cause when you fill the tank with water, the tank, the water as a fluid will take the shape of its tank. So what's the radius of the water that has this conical shape right here? Well, again, we could make a similar triangle argument. So the first one I want you to think about is that 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 gonna take half of the triangle. This is an example of an Asosceles triangle and we're gonna cut the Asosceles 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, this side length is gonna be the height of that triangle which is 10 meters. This side length is gonna be the radius which is four meters. 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 and again, we're just gonna take half of it. Pull this over here to the side. We get a triangle that looks like this. The height is gonna 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. These triangles are similar to each other so we can set up proportions. So this tells us that R over four corresponds to eight over 10 for which we can times both sides by four. This would give us that R equals eight times four over 10 which is gonna give you 32 over 10 which gives you 3.2 meters as the radius of the water at this point in the tank. And we could do this for greater generality, right? We don't just have to stop right here. What if the height is unknown? What if the height is say X meters at any moment in time? Well, we can set up these exact same proportions like we did a moment ago because what's so special about the number, what's so special about the number eight right here? Nothing really much. If we just said like, hey, what if the height is X? Well, then all of the subsequent eights have to get taken away. So you end up with a four X right here in which case then the general solution, let's start again in three, two, one. Suppose we wanna 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 gonna be 10 minus X, whatever the gap turns out to be. And so that then leads to the proportion that R over four corresponds to the height which is 10 minus X all over 10. In which case then we can solve for R easy enough. We get R is equal to four times 10 minus X all over 10, four and 10 both have a common factor of two which you can cancel out. And so we see that the radius is equal to two fifths times 10 minus X, which we can then check our answer. If we take X to be two like we saw before that would look like two fifths take 10 minus two which is gonna be eight, two over five times eight, two times two, excuse me, two times eight is 16 over five which 16 over five 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, why similar triangles can be very important in the calculus setting. Calculus cares about when we 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 dependent on the height of the water or in this case the gap of the water. And it's always, this formula 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 wanna talk about it in this trigonometry class.