Grades 11-12 Video Solutions 2025-2025_11-12

2025_11-12_26

Video Transcription

This is problem 26 on the grades 11 and 12, 2025 Math Kangaroo Contest. The diagram shows a regular hexagon ABCDEF. Point P lies on BC so that the area of triangle PEF is 64  and the area of triangle PDE is 42. What is the area of triangle APF? Our answers are A, 53, B, 54, C, 56, D, 60, and E, 64.  All right. So there are maybe a few ways to go about doing this analytically and with line geometry. And that is always an option.  If you feel comfortable with that, absolutely go for it. But in my opinion, it might get a little bit hairy and it will definitely get hairy kind of quickly.  So instead of trying to get it directly, maybe we could try to do some complementary counting and subtract off all of the other areas to find just the remaining area of APF.  Well, okay, how can we do that? Well, let's start with the area of the whole hexagon. So whole hexagon. Well, this regular hexagon is composed of like six equilateral triangles  and I'll draw in just one of them.  Sort of go to the center and then go to one of the edges. And what we notice is that we have this base here  and we can just call this S because it's like the side length of the hexagon. And we have the area of this equilateral triangle. That's a terrible equilateral triangle.  Right here, yeah, this equilateral triangle with side length S should have half the area of like this triangle PEF. And I say this because it has exactly half the height.  So if this is H1, this we can just draw this like repeated thing. This is also H1. So then the height of the entire triangle of PEF is H2 equals 2 times H1.  Awesome. Okay. So if a single equilateral triangle has half the area of PEF, that means it has area 32.  And therefore the area of the whole hexagon is going to be 6 times 32 equals 192. That's awesome. We like that. Okay. So now we have that.  And we have the area of like each of these things. We have the area of this triangle, like we are given this.  And we have the area of this triangle, another thing we are just straight up given. Okay. So now we care about these two triangles that I'm going to color in yellow.  And again, depending on the positioning of P, this might get kind of hairy. But what we can do instead is consider the area of them just together.  If we come up with maybe an expression, let's call this like distance X and call this S minus X. This also has height H1. So if we take the area of triangle A, B, P,  it should be S minus X times H1 all over 2. The area of triangle P, C, D is equal to X times H1, because again they have the same height, H1 all over 2.  And if we add these together, we get that it's actually S times H1 all over 2. And if we notice, that's also exactly the area of one equilateral triangle, S times H1 all over 2.  So then this just has the area of one equilateral triangle, which is 32. So this yellow section has area 32. This black section has area 64 plus 42, which is 106.  And the area of the whole thing has 192. So then we just subtract off 192 minus 32 minus 106 equals, that should be 54. So our answer then is 54.

Video Summary

The problem involves a regular hexagon where specific areas of triangles are given, and we need to find the area of another triangle. The transcript describes a method using complementary counting to subtract known triangle areas from the total area of the hexagon to find the area of triangle APF. The hexagon is divided into equilateral triangles, each with an area of 32. The known areas of triangles PEF and PDE are 64 and 42, respectively. Subtracting these from the hexagon's total area of 192 leaves the area of triangle APF as 54, which corresponds to option B.