Can You Find The Area Math Puzzle

Hello my lovelies, it's Susanna and today I want to show you how to solve this puzzle. A square is divided into four areas by drawing lines from the midpoints of its sides to one point inside the square. We can see the square here and from this midpoint of this side we draw a line to a point inside the square. We do the same from this midpoint of this side, drawing the line to the same point inside here. The same from this midpoint here and here as well. We are given the sizes of these individual areas and the question is how large is the red area? So we have to find the size of this area here. Okay, because it is a square we know that this side here has the same length as this side here. We know that these are the midpoints of these sides, so this part here has the same length as this part here. But because all sides are of equal length, we also know that these parts here, so this part has the same length as this part here. Okay, this is all the information we have, so let's highlight all these areas in different colors. We keep in mind that we have these sizes of these areas and we want to find the size of this area. Okay, so maybe we start with this green area here. What do we know about it? We know that it is of size 32 square meters and we know that this side here has the same length as this side here. But working with quadrilaterals is not so easy actually, so I would prefer to divide this quadrilateral into two triangles because working with triangles is way easier. And we can divide this into two triangles, for example like this. Then we know a whole lot about this triangle because we know that this side has the same length as this side, but the problem is that we don't know a whole lot about this triangle here. So maybe dividing it like this is not the best solution, but we could also divide this quadrilateral like this into two triangles because now we know something about this triangle here that this side has the same length as this side of this triangle here. And let's do that with all the quadrilaterals. So from this point here to this point here, I have two triangles now. The same thing here and here as well. So now we only have triangles and we are still interested in this area here, but now we can work with the area of a triangle because for a triangle there is a formula how to find the area. It is base times height over two and as a base I can just pick any sides of my triangle. So let's say this is my base and the height is always perpendicular to my base and passes through the opposite point here. So this is perpendicular, passes through this point, so this is my height. Now the thing is that the height can be inside of a triangle or if your triangle looks like this, for example, and you take this as your base, where is my height here? I have to be perpendicular to my base but I have to pass through this opposite point. So from the opposite point perpendicular down to my base here, this would be my height and it would be outside of my triangle. It's nothing bad but just keep in mind that both versions are possible. Either your height is inside or outside. But let's see what we have here. Maybe we start with this triangle. If I take this here as a base, I know that this part here is of the same length, right? Because this was a midpoint. So I have my base B here again in this triangle. So maybe we compare these two triangles. So I highlighted them here, we have our base here, our base here. What about the height of this triangle? Well, it has to go through the opposite point and goes down perpendicular. So this is my height of my blue triangle here. And what is the height of this red triangle? It has to pass through the opposite point here and has to go down perpendicular to the base. So this height is the height of the blue triangle inside the blue triangle. But this height is also the height of the red triangle but it just lies outside of this triangle. Which means that these two triangles have the same base and they share the same height. What does this mean for the areas of these two triangles? If they have the same base and the same height, they're going to have the same area. So if I call this area A, then I know that this triangle has the same area A as well. And if we compare these two triangles, let's see if that's the same case here. I take this here as my base. I know that this part here is of the same length because this was the midpoint. So I have my same base here. Now my height of this blue triangle, it's perpendicular to my base and passes through the opposite point here. So this is my height. And what is my height of my yellow triangle? It has to be perpendicular to this base and has to pass through this opposite point. So it's the same height. Also these two triangles have the same base and share the same height. So the areas are the same. So if I call this area B and this area B, it should not be the base. So I delete the rest. So these are just the areas. Same thing with these two triangles. Same way of explaining this. So I call this area C and this area C here as well. And the same thing for these two last areas. They share the same base and the same height. So they have the same areas. So in total we found A, B, C, and D areas. We have A twice, B twice, C twice, and D twice. And if we combine for example now the blue area here with A and B, and the green area with C and D, we know the sizes of these areas. So let's do that. We take A plus B plus C plus D and we know that it is of size 16 plus 32 which equals 48. So A, B, C, and D together are of size 48. We are only interested in A plus D. So only A plus D is what we need. But the great thing is that we have B plus C here. So we know this yellow area is of size 20. So B plus C is 20. So we have A here plus the 20 plus the D and this equals 48. We only need A plus D here. So let's get rid of the 20 by subtracting the 20 on both sides of the equation. Here we only have A. This cancels out. We have plus D and 48 minus 20 equals 28. And this is the area we were looking for. If you liked my video, please give it a thumbs up. It helps me a lot. I wish you a wonderful day and I hope to see you in one of my next videos. Take care!