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Challenge problems: perimeter & area
Lets do some example problems here, so we have the perimeter of each of the outer triangles is 30. So for example if I took The sum of this side, this side, and that side I will get 30 and that is true of all these outer triangles, these 5 outer triangles. They then tell us that the perimeter of FGHIJ So FGHIJ the perimeter of this pentagon right over here is 50 So if I add up that side plus that side plus that side plus that side, plus that side, I get 50. And then they say what is the perimeter of the star? So the perimeter of the star is really the outsides. if you take the bases away of each of these triangles. So it is this side, let me do this in a new color actually So the perimeter of the triangle i will do in orange. It is going to be this plus that plus that plus that plus that plus that i think you get the idea plus that plus that plus that plus that So the perimeter of the of the star so let me call this: perimeter perimeter of the star it is going to be equal to the perimeter of the 5 triangles is equal to perimeter of 5 outer triangles. Just call them 5 triangles like this minus their basis, right, if i take the perimeter of all of these sides If i added up the part that should not be part of the perimeter of the star would be this part,that part, that part,that part, that part and that part. those are not the part, those are not the part of the perimeter of the star so should be the perimeter of the 5 triangles minus the links of their bases links of their 5 bases. So what is the perimeter of the 5 triangles? well, the perimeter of each of them is 30, perimeter of 5 of them is going to be 5 times 30 which is 150, now we want to subtract out the links of their 5 bases now the links of their 5 bases if we add them up is the exact perimeter of this in inner pentagon right over here. So this inner pentagon has a perimeter 50, that is the sum of the 5 bases. So that right over here is 50, so the perimeter of the star is going to be 150 minus 50, or or 100. All we need is to get the perimeter of all triangles, subtracted out these bases which was the perimeter of the inner pentagon and we are done. Now lets do the next problem. What is the area of this this quadrilateral, something that has 4 sides of ABCD? And this is a little bit we have not seen a figure quite like this just yet, it on the right hand side looks like a rectangle, and on the left hand side looks like a triangle and this is actually trapezoid, but we can actually as you could imagine the way we figure out the area of several triangles splitting it up into pieces we can recognise. And the most obvious thing to do here is started A and just drop a rock drop an altitude right over here, and so this line right over here is going to hit at 90 degrees and we could call this point E. And what is interesting here is we can split this up into something we recognize a rectangle and a right triangle. But you might say how do, how do we figure out what these you know we have this side and that side, so we can figure out the area of this rectangle pretty straight forwardly. But how would we, how would we figure out the area of this triangle? Well if this side is 6 then that means that this that EC is also going to be 6. If AB is 6, notice we have a rectangle right over her, opposite side of a rectangle are equal. So if AB equals 6, implies that EC is equal to 6, EC is equal to 6, so EC is equal to 6 and if EC is equal to 6 then that tells us that DE is going to be 3. DE is going to be 3, this distance right over here is going to be 3. And we know that because if this is 6, this has to be something that we add to 6 to get 9, 9 was the length of this entire, of the entire base of this figure right over here. 9 was this entire distance so 9 minus 6 gives us 3, and now we have all the information that we need to figure out the area. The area of this part right over here of this rectangle is just going to be 6 times 7, so is going to be equal to 42 plus the area of this triangle right over here. Plus the area of this triangle right over here, and that is one half base times height one half. The base over her is 3, one half times 3 and the height over here is once again going to be 7 this is a rectangle, opposite sides are equal, so if this is 7, this is also going to be 7 one half times 3 times 7, so it is going be 42, lets see. 3 times 7 is 21, 21 divided by 2 is 10.5, 10.5 so this is going to be equal to 52.5 52.5 is the area of this entire figure. Lets do one more. So here I have a bizarre looking, a bizarre looking shape, and we need to figure out its perimeter. And it it first seems very daunting because they have only given us this side and this side and they have only given us this side right over here. And one thing that we are allowed to assume in this and you don't always have to make you can't always make that assumption and I just didn't draw it here I had time because it would had really crowded out this this diagram. Is it all of the angles in this diagrams are right angles,so i could have drawn a right angle here a right angle here, a right angle there, right angle there, but as you can see it kind of makes things a little bit, it makes things a little bit messy. But how do we figure out the perimeter if we don't know these little distances, if we don't know these little distances here. And the secret here is to kind of shift the sides because all we want to care about is the sum of the sides of the sides. So what I will do is a little exercise in shifting the sides. So this side over here I am going to shift it and put it right up there, then this side right over here, this length right over here I am going to shift and put it right over there. Then let me keep using different colors, and then this side right over here I am going to shift it and put it right up here. Then finally Iam going to have this side right over here, I can shift it and put it right over there and I think you see what is going on right now. Now all of these sides combined are going to be the same as this side kind of building, even you know this thing was not a rectangle,its its perimeter is going to be a little bit interesting. All we have to think about is this 2 right over here, now lets think about all of these sides that is going up and down. So this side i can shift it all the way to the right and go right over here. Let me make it clear all inside goes all the way to the end, right that it is the exact same all insde. Now this white side I can shift all the way to the right over there, then this green side I can shift right over there and then I have, and then I can shift, and then i can shift this. Actually let me not shift that green side yet, let me just leave that green side so I have not, I have not done anything yet, let me be clear I have not done anything yet with that and that I have not shift them over and let me take this side right over here and shift it over. So let me take this entire thing and shift over there and shift it over there. So before I count these two pieces right over here and we know that each have length 2 this 90 degrees angle, so this has link to and this has link to. Before I count these two pieces, I shifted everything else so I was able to form a rectangle. So at least counting everything else I have 7 plus 6, so lets see 7 plus 6 all of these combined are also going to be 7, plus 7, and all of these characters combined are all also going to be 6, plus 6, and then finally I have this 2, right here that I have not counted before, this 2, plus this 2, plus this 2. And then we have our perimeter, so what is this giving us, 7 plus 6 is 13, plus 7 is 20, plus 6 is 26, plus 4 more is equal to 30. And we are done.