Scale factors and area

- [Instructor] We're told that Polygon Q is a scaled copy of Polygon P using a scale factor of one half. Polygon Q's area is what fraction of Polygon P's area? Pause this video and see if you can figure that out. Alright, my brain wants to make this a little bit tangible and once we get some practice, you might be able to do it without drawing pictures but they're saying some arbitrary Polygon Q and P so let's just make a simple one. Since we're talking about area, I like to deal with rectangles since it's easy to think about areas of rectangles. And so, let's see, Polygon Q is a scaled copy of Polygon P. So let's start with Polygon P. And I will do this in red. So Polygon P, let's just say, I'm just gonna create an arbitrary polygon. So let's say that this is, and I'm gonna scale it by one half so I'm gonna make its sides have nice even numbers. So let's say this side right over here is four and this side right over here is equal to eight, this is Polygon P right over here, it's a quadrilateral, it's in fact a rectangle and its area is just going to be four times eight which is 32. Now let's create Polygon Q, and remember, Polygon Q is a scaled copy of P using a scale factor of one half. So we're gonna scale it by one half. So instead of this side being four, it's going to be two and instead of this side over here being eight, the corresponding side in the scaled version is going to be four. So there you go, we've scaled it by one half, and now what is our area going to be? Well our area, and this Polygon Q, and so our area is going to be two times four which is equal to eight. So notice that Polygon Q's area is one fourth of Polygon P's area and that makes sense because when you scale the dimensions of the Polygon by one half, the area is going to change by the square of that. One half squared is one fourth and so the area has been changed by a factor of one fourth or another way to answer this question, Polygon Q's area is what fraction of Polygon P's area? Well it's going to be one fourth of Polygon P's area. And the big takeaway here is if you scale something, if you scale the sides of a figure by one half each, then the area is going to be the square of that and so one half squared is one over four. If it was scaled by one third, then the area would be scaled, or the area would be one ninth. If it was scaled by a factor of two, then our area would have grown by a factor of four. Let's do another example. Here we're told, Rectangle N has an area of five square units. Let me do this in a different color. So Rectangle N has an area of five square units. James drew a scaled version of Rectangle N and labeled it Rectangle P. So they have that right over here. This is a scaled version of Rectangle N. What scale factor did James use to go from Rectangle N to Rectangle P? So let's think about it. They gave us Rectangle P right over here, and let's think about its dimensions. This height is one, two, three, four, five, it's five high, and it is, one, two, three, four, five, six, seven, eight, nine wide, and so its area is equal to 45. Now, Rectangle N had an area of five square units. So our area, let me write this down, so N area to P area, N area to P area, we are multiplying by a factor of nine. If we're going from an area of five square units to 45 square units, you notice N area is five, N's area is five square units, P's area, we just figured out is 45 square units, and so we have it growing by a factor of nine. Now what would be the scale factor if our area grew by a factor of nine? Well we just talked about the idea that area will grow, the factor with which area grows is the square of the scale factor. So one way to think about it is scale factor, scale factor squared is going to be equal to nine, or another way to think about it, our scale factor is going to be equal to three to go from N to P. Now let's verify that we answered their question but I just want us to feel good about it. Let's draw a rectangle that is scaled down from P by a factor of three. So a rectangle if we were to scale it up by a factor of three, we get rectangle P. So its bottom would have length three instead of nine. So it'd be like this. So that would be three, and its height instead of being five, it would be five thirds. Five thirds is one and two thirds, so it'd go about that high, it would look something like that. It would be five thirds. And so our Rectangle N would look like this, and what is its area? Well five thirds times three is indeed five square units. So notice, when we have the area growing by a factor of nine, the scale factor of the size to go from five thirds to five, you multiply by three. To go from three to nine, you multiply by three.