Number 22. The side length of the square shown in the diagram is 10 centimeters. The line down in the middle of the square divides it into two equal rectangles. What is the area of the shaded region? Pause if you want to try and solve it. Okay, so notice that the shaded region is made up of two of these triangles. And that means that if we can find the area of one of these triangles, we can just double it and then get the area of the whole shaded region. So remember that the area of a triangle is base times height divided by 2. All right, base times height divided by 2. So what's the base and what's the height? Well, it's a little bit difficult to tell because of the way that the triangle is oriented. But let's imagine turning this triangle like this and we'll have something that looks like this. It's a bit of a crude drawing, but we'll have a triangle that looks like that, right? And this side over here is this over here. And, you know, this side is this side. And then we have the long side over here. So what is the base and what is the height? Well, we can see that this over here is our base. And what is this length? Well, we have this nice diagonal. And remember, a diagonal, these two diagonals go through the center of the square, which means that this length over here is exactly half of this total length, right? And that length, of course, is 10. It's equal to the side length, which is 10. So it's half of that, so it's 5. That's 5, which means that this over here is 5. So our base is 5. All right, let me clear this a little bit. What about our height? Well, our height, you can see in this diagram over here, is our height is this length over here. It's actually outside of the triangle, but that's going to be our height. If we extend this side, then this over here is going to be our height. And that's equal to this part over here, right? If we extend our base over here, it's this line. So our height is this length. And we can see, because this middle line divides it equally in two, that this is going to be 5. This is going to be half of the side length, which is 10. Half of 10 is 5. So this is also 5. So now we have our base. Now we have our height. So it's 5 times 5 divided by 2. But remember, we actually have two of these triangles, right? So we have to multiply it by 2 because we want the sum of both of them. And divided by 2 times 2, that cancels out. So it's just equal to 5 times 5, which is 25. So that is answer option B.

shaded region

square

triangle area

geometry

mathematics