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Pick's theorem says that the area of a simple polygon $P$ in the plane with vertices that are all lattice points (that is, points with integer coordinates) equ...

Question

Pick's theorem says that the area of a simple polygon $P$ in the plane with vertices that are all lattice points (that is, points with integer coordinates) equals $I(P)+$ $B(P) / 2-1$, where $I(P)$ and $B(P)$ are the number of lattice points in the interior of $P$ and on the boundary of $P$, respectively. Use strong induction on the number of vertices of $P$ to prove Pick's theorem. [Hint: For the basis step, first prove the theorem for rectangles, then for right triangles, and finally

Pick's theorem says that the area of a simple polygon $P$ in the plane with vertices that are all lattice points (that is, points with integer coordinates) equals $I(P)+$ $B(P) / 2-1$, where $I(P)$ and $B(P)$ are the number of lattice points in the interior of $P$ and on the boundary of $P$, respectively. Use strong induction on the number of vertices of $P$ to prove Pick's theorem. [Hint: For the basis step, first prove the theorem for rectangles, then for right triangles, and finally for all triangles by noting that the area of a triangle is the area of a larger rectangle containing it with the areas of at most three triangles subtracted. For the inductive step, take advantage of Lemma $1 .]$



Answers

(a) If $ C $ is the line segment connecting the point $ (x_1, y_1) $ to the point $ (x_2, y_2) $, show that $$ \int_C x \, dy - y \, dx = x_1 y_2 - x_2 y_1 $$

(b) If the vertices of a polygon, in counterclockwise order, are $ (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) $, show that the area of the polygon is
$ A = \frac{1}{2} \bigl[ (x_1y_2 - x_2y_1) + (x_2y_3 - x_3y_2) + \cdots + (x_{n-1} y_n - x_ny_{n-1}) + (x_ny_1 - x_1y_n) \bigr] $

(c) Find the area of the pentagon with vertices $ (0, 0) $, $ (2, 1) $, $ (1, 3) $, $ (0, 2) $, and $ (-1, 1) $.

Okay. Today I'm going to show you how to use the It's lie integral and the green searing. So first thing it says and then we use the tea instep X and Y two funds. Ah, we write a we write x and y into t sing Then we can use t and then we can calculate the X n d t. So this one is the line integral here. So first thing, first thing here So we knew we can try x equal to x one plus t x two minus one When he is thesis ty's c 021 and the X can be feet And why equal to this They also can be feet and then we we can get the x d y if you don't know how to calculate the x t y. So you just write a d y he he here So the t y did he just equal to finest you? Why? To minus t y one And you moved to this want you here you get these. So so then we we get teeth. Then we just use the accent. Why use TV instead of accent? Why and did he instead of the accident. Ey here. So this fly integral is become to just one variable integral And in here, x Q and X one are constant. Why? I went to sell Constant After we simplify them and we calculate we get this okay for part B, we need to use the Massey off Messi off party to do the party here. So we also can write this thing here x equal to this And why equal to this is similar with stuff are a and he also from 0 to 1. But this one is for every every piece line integral for example they have like something here. So this one called C one and this one is C two and C three. Here's the eyes. The start point I plus one is the end point for C one. We're beginning from ex CEO and we end on X one everything. Let me clarify everything, make you understand So then we cooking dx and dy y here. So then we use the green serum to calculate their sings is from part a We maybe can guess for probably maybe we try to use the part a expression to calculate party. So to consider the area we think about the green Syria. Right. So this one is the green Syrian here, and this one is equal to lie integral. So after we calculated curl and we get you So since here is too and we interviewed that we just get two times area and we equal to the my integral on right hand side. Then we just use t two instead of everything here. Now. Here. Sorry. So this one is the result. We divide by two on here. So this is the one half come from then. We just need to program in this formula X one x two x Wait. I think here should be X. Okay, it's X Q. My bad reading. Sorry about that. So we just need to program this formula which point Which point? And here and we get Just use the area. Thank you.

This video were asked to reversible over Steve Extra Light minus y X is equipped next one way to my next one worth see as the line segment connecting X one y one to extradite. So, first of all, with a for amateurs display So we're going to subtract extra my 61 and why to minus Why one when we get our gaze in her are you gonna be? And we know that X is x one plus a B and bodies by one bt so excess extra horse bt and vines. Why one horse beauty services That's what uh so X is a time Steve's or a is extra bling. That's what and why is why one plus R B. I'm just on our t varies from 0 to 1. So if your partner zero for tea, you're gonna get X one y one, which is your first point, and you you plug in 1 40 you're going to get next to Los X 2.6 just next to and why is he put the next one plus extra Linus? Sorry. Why why is he But why? I love plus y tu minus one. I just want to So we get next to come off like So this is our perimeter organization. So now we're gonna change the integral so you can change this integral right here. All in terms of So we're going to We're gonna We're gonna try to take me to go over perspective. So first we have the integral from 0 to 1 on our exes exploit Plus expired next to mine. Sex won t on our he wise. Just a derivative off. Why would respected t Just just want to remind us why I wanted to You minus now are why is why one last point to a minus Bi won t times DX, which is just still buying sex works. But I weeks bad So we have X one times like two lines. Why one plus extreme wings for one for times like to minus 41 times T minus one times extra one sex one minus one to minus five extra money sex 70 on all that. So this might look a bit tricky, but it's pretty straightforward. You're just distributed that we noticed that this this are equal so we can still against hero and then we're gonna distribute are explored. We're going to distribute. So we get back sometimes. Like to buy this. Excellent. So I want minus y one times, X two plus by one time sex for here again, we notice that we can cancel our X one y one. So we're left with being truthful from Syria. One of X one times one to minus six to minus 41. The teeth. And since we have no t here, we've been uncovered as a constant and the integral of the cheese. Just to you. S t goes 101 So you have excellent times. Like to my sex, dear line, the extra times for everyone and one linings. Here's just for him, sir. Now we proved what we wanted to show that integral it's equal to X one. Why? To my business too. She's exactly this is exactly what I wanted to show. Only just did. All right. Great. So far number report be They're telling us in general, we want to show that the area this report true 1/2 x one times right. Two lanes 62 times one suspects three times life or a minus Uh uh X three times by two. And but what stopped the doctor? Plus excellent backs renderings One times Laurean Mining sex in terms right in this one so over and have to prove that person you know is that the area beginnings half the close in a row for the integral of X t y minus. And for a general point that state starts the next morning goes to x. Sorry x I Why I enclosed to expire plus one y plus one. But we want a first fewest. Her amateurs. So the thing is, you take x I plus one minus x side. That's why I plus one minus y I said, This is are a of the and we know that back says X one horse painting and white one plus beat so are excellent is just excited about a expect plus one minus X. I we want to play that by I know why it's right one. Just why I plus why I lost board, find this white guy Times and t goes from zero to okay. Now you need to determine the extent deities with Texas just excited. What's one less exciting? Andi lies just why I plus one minus y And now again, just like importante We cook air for things back into musical on We just break from zero and I'm after break Distribute. You noticed that we can do some cancellations. You can see this with this. And then all what we're left with is and and then we're simply been there the first term and the third term district. So we have 1/2 on the outside, Mrs. One formula for area ventured Muslims a little one. No backside like I post one minus exciting by I minus Y I x nine Force one Plus Why I times excited again. You can cancel. That's with this. So I just left with 1/2 times in to go from 0 to 1 of Rex started y plus one minus x I plus born. Why I And then since these are all X's and y's and don't contain to use, we can force them to the outside and the integral DTs just cheese was 1/2 excited. Why? I posed by an expects are one by all right, and then t as it goes from 1 to 8 from 0 to 1. What? This is just one minus zero Or just so now we get that the area is 1/2 excited. Why? I close one in this ex cycles. One point then, if we want to find the total area or what we do is reach some from I through end minus one 1/2 excited. Why bust one minus X I plus one? Why I and then we get what we wanted to show. But there's despite made a little bakery in here, so it should be X ready. Why now that now we proved what we wanted to show and now for part, really have to do is we have a Pentagon to determine the area from the former lovers Start. We found a part dean and were given 12345 coordinates and just slave about explode y one x 212313 Forward for expert for five. And then now you're gonna put these into the formula of returning in Part seven for feeding Germans before and now it's just straight up again. Chuck, there's nothing more. So we have 1/2 times zero minus zero times weren't buying. Sierra Times two plus two times three minus one times one plus one minus two zero minus three 10 times one, uh, lying us the negative one times two. So basically what we can formula right here, right here and then for end. We just put it five. So that's that's That's our value for Ed. And then when we plug in the values relabel the except Flex X is in the lives what we get. And nine, perhaps 4.5.

So we want you. Meaning my the area off this jungle here. Minimized area here. Andi, listen. We go Jew. Meaning my, uh, half of extends. Why, Andi? We see that the X and Y has the relationship between, um, badges in the similar change laws now. So we have to jangles. The 1st 1 will be in rat here and the other one will be in in blue here. So we see that that we see ANC's off x minus. I were ico Jew on to why of ah be And then on why can be written is they be ex dividing by ex minister So we can have one job. Do you find a function and ex 1/2 on the X times with the B ANC's of ah x minus. I saw it governed as is one of 1/2 ah, be outsize or ex grand dividing by act minus I So every prime X we go Do we had we had to do? Yeah, we have ah x minus any square and inside Which again, huh? On the terms again The Jew X times X minus I minus X square. So he sends an acre Jizo means done. Which would, uh, new Morita, a good Jewish zero. So that means none. Which again, huh? Do X square minus to my angst minus x squared. It could Jizo so means done. Which again, huh? Ah two will be X square minus. Do I X a coach's also X. I'm signed them have X minus two. I could just go. So doesn't implies that x must echo Ju ju. I know this really here must echo due to die. And then why were Doesn't implies them why we called you. Ah, be times too. I divide by two ominous. I would be I. Then we're gonna to be so And then the stronger echoed you? No, Why? Remember x so it could choose to be miners. And we have the slum going down so have too good a man as near to be over too high. So it could you manage being a and really that the bone here will be under meet point Because you have accurate Did you I And while could you two be examines and weak point

We know that we know angles with pairs of parallel sides re Furthermore, no. The A s air angle side angle era which states that the side in between two angles has to be congruent. Therefore, because we know that certain segments and congruent triangles, group assignments and congruent triangles must be congratulated because their corresponding we know that a X has to be equivalent to be why, okay, we know that if two lines intersecting to probable planes or parallel the respective distances between the planes in the direction of the lines, will be parallel. So like if we've got two lines like this, we can see the segments of the two parallel lines crossing the two parallel lines. Parts of the two parallel planes have to be equal.


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