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let's say we've got a rectangle draw the rectangle there's the rectangle and we have two diagonals across the rectangle that's one of them and then we have the other diagonal and this rectangle has a height of H so that distance right there is H and it has a width of W it has a width of W what we're going to show in this video is that all of these four triangles have the same area now right when you look at it it might be reasonably obvious that this bottom triangle this bottom triangle right here that this bottom triangle right here will have the same area as the top triangle as this top kind of upside-down triangle that these two have the same area that might be reasonably obvious they have the same dimension for their base this width and they have the same height because this distance right here this distance right here is exactly half of the height of the rectangle they are symmetric they are equal triangles they have the same proportions now it's probably equally obvious that this triangle on the left that this triangle on the left has the same area as this triangle on the right right here that's probably equally obvious what is not obvious is that these orange triangles have the same area as these green blue triangles and that's what we're going to show right here so all we have to do is really calculate the areas of the different triangle so let's do the orange triangles first before doing that let's just remind ourselves what the area of a triangle is area of a triangle is equal to 1/2 times the base of the triangle times the height of the triangle that's just basic geometry now with that said let's figure out the area of the orange triangle the area of the orange triangle it's going to be 1/2 times the base so the base of the orange triangle is this distance right here it is W it is w so 1/2 times W I'll do that in 1/2 times W I wanted different color in the color I wrote the W in 1/2 times W now what's the height here what's this height what's this distance well we already talked about it it's exactly halfway up the height of the rectangle - exactly so times 1/2 times the height of the rectangle times 1/2 times the height of the rectangle so what's that going to be you have 1/2 times 1/2 is 1/4 times width times height so 1/4 width times height so the area of that triangle is 1/4 with height so is that one same exact argument they have equal area now what's the area of these green or these green blue triangles well once again area I'm gonna do this in a green color do this in the area is equal to 1/2 base so these guys are turned on their side the best base I can think of is this distance right here that distance right there if you look at this triangle it's this distance right here it is the height of the rectangle so now we're dealing with the base in this case is the height of the rectangle don't want you to get too confused the height is now going to be what the height of this triangle so these triangles are turn on the side so what is this distance right here what is this distance well it is exactly half of the width right we're going exactly half of this distance right here this point right here is exactly halfway between these two sides and halfway between those two sides so this distance right here is 1/2 the width or the height of these sideways triangles are one-half of the width so the height here is one-half of the width a little confusing the base is equal to the height of the rectangle the height is equal to one-half of the width but if you do the math here the area is equal to 1/2 times 1/2 which is 1/4 height times width or you could just write that as 1/4 width times height which is the exact same area so the area here is 1/4 width times height area here is 1/4 width times height which is exact which is the exact same area each of these orange triangles and it makes sense because each of them are exactly one-fourth the area of the rectangle hopefully you enjoyed that