Hi there. This is Ryan Malloy here at the Worldwide Center of Mathematics. In this video, we are going to answer the following question what is the relationship between the ratio and scale factor between two triangles. So here we have triangles, blue and yellow and we're told that they are similar, meaning their corresponding angles are congruent. But that doesn't tell us anything about their sides. This one might have very different side lengths than this one, similarity only refers to the angles. So let's say that this side here has a length of L and suppose that we also know that the area of this triangle is A where A now can be any number. The question is what happens to the area if we double the side length and what you might think is that the answer is simply the area is doubled, side length doubled, area doubled. Seems simple enough, but it's not quite true. The area is proportional to the side length squared, not just the side length. So if we double our side length and our proportionality statement, what we get is not 2L but 4L squared. So naturally the area should be quadrupled, 4L, 4A rather which may be a somewhat counterintuitive result but it may make a bit more sense if you consider it in the context of a square for example. Since square is just base times height if you double one, the area will multiply by 4, not by 2. Okay well what about the perimeter of the triangle? This has a total perimeter of P and we want to know what is the perimeter of this triangle if all the side lengths are doubled. Well we might think, oh, well we just did our proportionality statement therefore, this must be 4P but you have to be a little bit more careful than that. So let's see here, if this is L, we'll call this a prime, a double prime. Well if this side length is doubled then this side length will just be doubled as well, 2L prime, 2L double prime. So if we have L plus L prime plus L double prime is equal to P. Now we have 2L plus 2L prime plus 2L double prime, well we can simply rewrite this as 2 times L plus L prime plus L double prime but we see right here that this is equal to P. So the whole thing just becomes 2P. And there we have it. My name is Ryan Malloy and we've just discussed the relationship between the ratio and scale factor between two triangles.