 Hi, this video is called similarity statement and scale factor one. If you look at this problem, there's actually three parts to it. They are asking us to write a similarity statement, find X, and the scale factor of the two polygons. So in this problem, we are going to be comparing two triangles. We're told that they're similar, so that means they're proportional, which means that all the side lengths grow or shrink at the same amount, and all the angle measures will be the same. When they ask you to write a similarity statement, it's actually a pretty simple thing to do. They just want you to formalize what vertices match in one triangle match with the vertices in the other. So you're allowed to name your first triangle whatever you want. In this case, I'm going to name it Jkl. The one squiggly mark means similar. And now when I name my second triangle, I have to make sure that the letters match up correctly. So for example, the J is in the lower left-hand corner. It looks like it's representing an angle that might be 90 degrees. Well, in my second triangle, R is the one in the lower left-hand corner, but more importantly, it's the one that looks like it's also 90 degrees. So since the J and the R match up, since I listed the J first, I have to list the R first. Then the K comes straight up from the J. So does the S. So the K and the S need to be in the same position. And then lastly, when you come down the hypotenuse, you end with L and you end with T. So L and T should be in the same position as well. So all the similarity statement does is match up the letters. J goes with R, K goes with S, L goes with T. So that helps you figure out which side lengths belong with each other. So we are already done with the first part of this problem. That is the similarity statement. Now let's work on finding X and finding the scale factor. And basically, we have to find the scale factor first to find X. What will happen is we just have to match up our side lengths correctly. Here's X. It matches up with 24. 25 matches up with 10. And 65 matches up with 26. So let's see if we can figure out how much this shape grows or shrinks by. I'm going to make three ratios or three fractions to help me figure this out. I've got 25 over 10. I've got 65 over 26. And I've got X over 24. Alright, so I have my three ratios set up. And right now they all look pretty different. 25 over 10, 65 over 26, and X over 24. But the truth is, since we're told that these triangles are similar, these fractions should actually all be equivalent and represent the same amount. So let's simplify. 25 divided by 10, I can divide out a 5 from the top and the bottom to give me 5 over 2. 65 over 26, I can divide 13 by the top and the bottom. That also gives me 5 over 2. So it looks like I just found my scale factor. 5 over 2 tells me how much the shape changes. So scale factor is going to be 5 over 2. Now if you set up your fractions and you reverse, you had the 10 on top, the 25 on the bottom, the 26 on top, the 65 on the bottom, that would give you a scale factor of 2 over 5. I personally, this is Mrs. Milton, would accept either answer for your scale factor. You might want to check in with your teacher to see if they want a specific answer and they can tell you what they would be looking for there. Alright, so we've got two of our three questions done. All I have left now is to figure out what X is. Well, I know my one side length pair changes by 5 over 2. I know my second side length pair changes by 5 over 2. That's how we got our scale factor. So I know my third side length pair should also change by 5 over 2. So I have to figure out what X is so it can simplify to 5 over 2. So this is actually just going to be a proportion problem where we'll cross multiply. You can take your X over 24 and set it equal to the scale factor of 5 over 2. So it'll be a simple cross multiplication problem. Let's see. We have 2 times X will become 2X and 24 times 5 ends up as 120. So when you divide both sides by 2, you get X equals 60. Sorry, I don't know what's going on with the smart board. That is a 6. Alright, so we've answered all three questions. We've got our similarity statement. We've got our scale factor and we find X. I'm just going to do one more thing to make sure we did this correctly. If I replace my X with a 60 and you end up with 60 over 24, I need to make sure that that reduces to 5 over 2 because the triangles are similar. All the side pairs need the same scale factor. So think about it. What could I divide out of a 60 and a 24? You could divide out a 6. 60 divided by 6 is 10. 24 divided by 6 is 4. So then when you have 10 over 4, you could divide by 2 again and sure enough you get 5 over 2. So you have confirmed that all three side length pairs have that same scale factor and that's necessary for congruent triangles. So we've gotten our three answers. You've gotten some really good practice on similarity statements, setting up proportions, matching things up correctly in similar triangles and finding a scale factor.