 7.1 and 7.2 are all about proportions and similar polygons. Make sure you get this into your notes. 7.1 and 7.2. So 7.1 is all about proportions. What is a proportion? In order to understand proportions, we need to understand ratios. A ratio. So the ratio of integers is the number A to B, or the fraction A divided by B. In other words, it's a way of comparing two values. So take for example, let's say our class had 18 males and 15 females. The ratio of boys to girls would be 18 over 15, which reduces to 6 to 5. If we turn that around and said what's the ratio of girls to boys, that would be 5 to 6. And if we asked for the ratio of girls to total students, that would be 5 to 11. So a proportion is when you have two ratios set equal to each other. So for example, 4 to 5 is equal to 12 to 15. 3 to 5 is equal to x to 75. So we'd have to maybe solve for x. And then in general form, we would say our proportion is A over B is equal to C over D. So to solve that middle proportion, you take a cross product. 5 times x is equal to 3 times 75, which simplifies to x is 45. So let's take a look at an example here. Here we have a triangle and the ratio of the measures of the three sides is 5 to 12 to 13. And the perimeter is 90, and we want to find the measure of the shortest side. So looking at that proportion, 5 to 12 to 13, that kind of tells us how different parts of each side are apportioned. So if we have a triangle, we know the short side contains five parts. The second longest side is 12 parts, and the longest side is 13 parts. So if we want to find the measure of the shortest side, we need to figure out how long is one part. Well, we know all the sides add up to 90 centimeters. So 5 parts plus 12 parts plus 13 parts equals 90. And solving that, we get that one part is equal to 3 centimeters. And so therefore, if we substitute P equals 3, we get that the short side here is 15 centimeters long. Take a minute and jot down these two proportions examples. Pause the video. Try them on your own. Did you pause the video? Good. With the first one, the cross product gives us 6 times the quantity 4x minus 5 equals 3 times negative 26. And solving gives us that x equals negative 2. With the other example, our cross product gives us x minus 3 times x minus 7 equals 8 times 12. Foiling out gives us x squared minus 10x plus 21 equals 96. And so then we'll have to solve by factoring. We should get a final answer of x is 10 or x is negative 5. Two possible answers here. One final example. This monument is in South Dakota. It's called Mount Rushmore. And these faces of presidents were actually carved into the side of the mountain. Here we have George Washington, Thomas Jefferson, Teddy Roosevelt, and Abraham Lincoln. If we take a closer look at Abe Lincoln. So on Mount Rushmore, Abe Lincoln's mouth is 18 feet wide. In other words, that length is 18 feet. If we use that same scale factor and his image were carved head to toe, in other words, his entire body were carved into the mountain. How tall would it be? We'll set up proportions. And so we need mouth in real life divided by mouth on the mountain is equal to height in real life divided by height on the mountain. Well, we know the mountain's mouth in that's 18 feet. We need to know the other parts. Well, Wikipedia search finds that Abe Lincoln's height is 76 inches. He was six feet four inches tall. And we can just kind of ballpark how wide his mouth would be. If you think about how wide your mouth might be, maybe it's let's say two and a half inches. If we say two and a half inches, then the height of Abe Lincoln where he'd be carved head to toe on the mountain would be 547.2 feet. That's a tenth of a mile tall. Section 7.2 is all about similar polygons. When we talk mathematically, the word similar means same shape and different size. And so similar polygons, we say the corresponding angles of those polygons are congruent and the corresponding sides are proportional. And so then two similar polygons have a scale factor, which is the ratio of the corresponding sides in similar polygons. I'll give you an example. If we say quadrilateral ABCD is similar to EFGH, then all the angles must be congruent, A congruent to E, B congruent to F, C congruent to G, D congruent to H. And all the sides are proportional. In other words, AB divided by EF must equal BC divided by FG. And that must equal CD divided by GH, and that must equal DA divided by HE. In this case, that is indeed true. 7 divided by 3.5 is equal to 10 divided by 5 is equal to 12 divided by 6 is equal to 6 divided by 3. So in order to show that two polygons are similar, we need to prove that the corresponding angles are congruent and the corresponding sides are proportional. So are these two triangles congruent? Are they similar? Well, we know that angle K is not congruent to angle P. And since these are both isosceles triangles, we would need for those angles to be congruent in order for anything else to be true. And so therefore they are neither congruent nor similar. How about these two triangles? Well, in triangle ABC, we know two of the angles, 60 degrees and 80 degrees. Using the triangle sum theorem, we know that angle C must be 40 degrees. Likewise in triangle RST, we know angle R is 60 degrees. So therefore we know that all the corresponding angles are congruent. Next, we need to check to see whether the sides are proportional. And so all three corresponding sides, AB must correspond with RS, BC must correspond with ST, and AC must correspond with RT. We need all three of those ratios to be equal. So 5.2 divided by 3.9. Is that equal to 7 divided by 5.25? And is that also equal to 8 divided by 6? Well, if you pull out your calculator, you see that all three of those equals 1.3 repeating. And so therefore the sides are proportional. And so we say that triangle ABC is similar to triangle RST. Here we know that the polygons are similar. Write a similarity statement and solve for X. So here we know that triangle WXY would be similar to STU. And we want to solve for X. Well, X, since we have two isosceles triangles, we only really need to solve for one of the X's. ST is that length. ST corresponds with WX. In order to create proportions, we need a known ratio. In other words, we need a ratio where both lengths are known. Well, we know 3 and 9 are corresponding sides. And so that will allow us to set up a proportion. WX over ST is equal to XY over TU. And so that means 4 over X plus 5 equals 3 over 9. And solving for X gives us that X equals 7. Here's another pair of similar polygons. The similarity statement here. If we say triangle ABC, that must correspond with triangle DEF. Now DEF is in a different orientation. So we'll have to take care of writing a proportion to solve for X. If I say AB, AB is the first two letters in triangle ABC, that must correspond with DEF. So DE are the first two letters. AB over DE equals AC for DF. Cross multiply to solve for X, and we get that X is 7. Here we have two pentagons that are similar. We need to solve for X and solve for Y. So we'll actually kind of solve two separate problems. First, let's write a similarity statement. If I say ABCDE, that would correspond with RSTUV. And we can see that by the congruence markings on those angles. So all five angle pairs are congruent. Next, we have our known ratio, 6 and 4. And that is the key to unlocking the rest of the variables. That'll tell us what our scale factor is. So if I want to solve for X, CD, I see that corresponds with TU. And so CD over TU would equal AB over RS. Again, I've got my known ratio and my unknown ratio. That helps me solve for X. Next, I need to solve for Y. And I see 8 and Y plus 1 correspond. And so I could say UV divided by DE must equal that same known ratio, RS and AB. So again, Y is 13 thirds or 4.33, repeating. Next, there's no picture with this one. We have two similar rectangles, WXYZ and PQRS. And they have a scale factor of 1 to 5. What that means is that any measurement in WXYZ, the same measurement in PQRS must be 5 times larger. So if I have a triangle, let's call this WXYZ, PQRS must be 5 times larger in every measurement. So if we have the length and width of PQRS, which is the larger of the two rectangles, they have 10 meters and 4 meters, we want to find the length and width of this smaller triangle. So if we could talk about the length, the length of measurements, the length of WXYZ divided by the length of PQRS would be 1 over 5. And so that means X has to be 2 meters. Likewise, the width, that ratio, must be 1 over 5 as well, 1.25 meters.