AULA: Geometria analítica: Equação da reta, parábola e círculo | RÁC. LÓGICO

Added: 2026-05-26

[00:00:00]It should have been already.

[00:00:01]Let's stream it and see if YouTube opens here, I'll be right back.

[00:00:04]That.

[00:00:06][snoring] Hello, hello. [sighing] We are starting the broadcast.

[00:00:16]Wait for the staff to enter the room.

[00:00:24]Bruno, I'm already there. You can tell me if the audio is low, if it's okay, if it's clear to you that we're just aligning the technical details here and we'll get right to the content, which is what matters most.

[00:00:48]But the technical aspects also have to be perfectly aligned, perfectly correct, because I confess that mathematics is not such a simple thing.

[00:01:00]For me it is, it wasn't at first, but liking and being passionate about the business made it simpler for me.

[00:01:17]How are we doing, Robson?

[00:01:34]Guys, if you can, please answer in the chat if this is clear to me.

[00:01:44]If the audio is clear, let's continue. We're already on our 10th meeting. We covered a lot of content.

[00:01:59]Geometry, that part.

[00:02:03]Oh, that's great, that's great, that's great, that's great.

[00:02:06]Geometry is always a bit more robust, isn't it? There's more to geometry, because we start with plane geometry, move on to spatial geometry, and now we're going to finish with analytical geometry.

[00:02:22]Just a reminder, folks, goodnight to everyone. Tamara, Robson, Bruno, whoever joins in, if you could say where you're from, we remind you that the content is broad, so the student also has to find their own ways to study, perhaps a detail that we don't have time to talk about here. Why?

[00:02:51]Because this is a live stream, it's a summary of our meeting. I call it a summary meeting where we gather information, and with my experience in competitive exams, what do I do? I'll take the topics, the points of theory that are most relevant to you, and bring them here in this quick back-and-forth and problem-solving session we have.

[00:03:22]There's analytic geometry, there's a book, folks, the elementary mathematics series, if anyone wants a recommendation for a series to specialize in basic mathematics, there's practically a whole book just on analytic geometry. So, it's a broad topic. What are we going to get together here? Papum, chew on whatever falls so you can earn your points and get your approval at Petrobras, which is the most important thing. Name in the official gazette and dream fulfilled.

[00:03:57]Without further ado, let's get down to the week's problems. Ah, the week is just beginning, even better, fewer problems. Just don't let the problems from last week accumulate. It's time for us to focus and bring to mind the reason why we study, why we're in this battle to achieve our approval, and leave everything else aside.

[00:04:26]Come with me.

[00:04:29]In analytical geometry, first you have to understand, you have to understand what?

[00:04:37]Analytic geometry comes from basic geometry, which is plane geometry, and then from spatial geometry.

[00:04:47]And it's kind of been called algebraic geometry. What do I want? That it's a geometry of position, right? Analytic geometry was developed there by mathematicians, one of whom is Descartes, Pier de Fermar.

[00:05:05]So what does he bring?

[00:05:08]A bridge, a way of relating basic geometry, plane geometry, to that object within space, its position within a space.

[00:05:25]Gentlemen, a simple but tricky question. Do you think an engineer—let's use a crazy job title here, NASA trajectory engineer —needs to know analytical geometry?

[00:05:45]For God's sake, if you input the wrong position or coordinates for the spacecraft's docking or reentry trajectory, people will die. It's not just us who die, right? The material is lost, the entire budget goes down the drain due to a miscalculation. So, there it is, so you know exactly how to position a space shuttle, the space station, that's pure analytical geometry, because it's a 3D space where you have coordinates to explain it using numbers. You explain, using numbers and the Cartesian plane, where this object is positioned.

[00:06:40]And in the world of gamers, is analytical geometry used? Furthermore, for God's sake. Here's a construction screen, one of those screens where you're developing the game, positioning graphic elements, depth, all that stuff.

[00:07:04]Uh, the engineer from the company, you know, from EA Games, Sony, PlayStation, I don't want to, I'm not going to advertise here, but I already did.

[00:07:14]The person has to know this very well, very, very, very well, understand why you need to learn and understand this in your life, in your day-to-day life. Oh, damn, will I be using this in my day-to-day work at Petrobras? Of course it will, of course it will. Depending on your role there, whether it's technical or whatever, you'll use analytical geometry. You definitely need to learn.

[00:07:46]Let's learn.

[00:07:50][sighing] What is the [snoring] basis of the positional system? He's talking about it over there. It's the Cartesian system, because it comes from "descartes orthogonal."

[00:08:04]Why is it orthogonal here?

[00:08:13]Perpendicular to each other. You can say perpendicular lines or you can say orthogonal lines. So, the Cartesian system, it's there, look, the traditional one that you learn in first-degree functions, that you learn in second-degree functions, second-degree equations, X-axis, Y-axis.

[00:08:40]X-axis, Y-axis. The quadrants are important. The quadrants are just a matter of bringing your knowledge of the trigonometric circle. I'm going to reiterate here for you who is in the first quadrant.

[00:09:00]This is the first quadrant.

[00:09:05]This is the second quadrant. This is the third quadrant. This is the fourth quadrant. What are the quadrants numbered?

[00:09:19]in a counter-clockwise direction. Extremely important.

[00:09:25]Now I'll make a point here.

[00:09:28]Let's score a point.

[00:09:32]Actually, it's already marked here, which is what? Any point that has coordinate X.

[00:09:45]Let's put it here on the side, see?

[00:09:51]X Y.

[00:09:55]If it's a point, if you name the point A, you put X of A, Y of A, X abscissa. How do you record what abscissa is? What is ordered?

[00:10:09]Phonetics helps you a lot. I recorded it like this, absissa. The abscissa is the x-axis, the ordinate is the horizontal axis, and the ordinate is the y-axis, the vertical axis. Completely shut down. All of these names are in the legend, OK? So, when I talk about the ordered pair, the last line there, this little 'a' here, is the abscissa.

[00:10:52]And this B is the y-coordinate of the given point, in this case, point P.

[00:11:05]What will they ask you? What are the first things you need to know?

[00:11:13]Remember, Cadu, I can bring knowledge from the plane and draw these plane figures there within the Cartesian plane. Yes, you can.

[00:11:30]Yes, it actually helps most of the time; plane geometry helps you, you understand? And plane geometry is what provides the basis for spatial and analytical geometry.

[00:11:43]So you have to be pretty much on the ball, don't forget. Distance between points. The first concept mentioned is distance.

[00:11:54]If you're designating point AB there, it's a positive distance.

[00:12:03]Here, the distance is expressed as if it were in absolute value.

[00:12:08]So let's understand where this formula came from, which is a general formula that will be used according to the coordinates of two points on a Cartesian plane.

[00:12:20]Let's go.

[00:12:22]Let's use this drawing here. Uh-huh.

[00:12:26]Giving it a boost.

[00:12:31]No, it turned out bad.

[00:12:36]Let's improve this business, okay?

[00:12:43]A, B. These are points A and B. So, these points have their coordinates on the plane, OK? This one here, look, is the X of A and the X of B.

[00:13:09]Here, Y [clearing throat] and Y of B. Every point has an abscissa and an ordinate. They are marked there.

[00:13:21]How do I know what the distance is? Just memorize the formula, you'll see. You can decorate the formula. What is the distance between points B?

[00:13:29]The square root of the difference of the squares of the x-coordinates.

[00:13:36]But it didn't come out perfectly here, but you can correct it here. This is just one more.

[00:13:44]The difference between the squares of the x-coordinates plus the difference between the squares of the y-coordinates. So XB is XA² plus YB - Y². This is all very fundamental. Where did this come from? I'm only going to demonstrate one formula, but what else do we have time to demonstrate? all the formulas.

[00:14:15]And it's not very productive for us to demonstrate all the formulas, because you won't be asking for a demonstration on the test. You need to understand some demonstrations now, and on the test you'll only need to memorize the application of the formulas. Completely shut down. Look how interesting.

[00:14:32]Analyze this triangle here.

[00:14:42]This right triangle here. Let's do the following. This distance is what I want.

[00:14:50]That distance is the hypotenuse of that little black triangle. So what do we need? Find the legs of a right triangle and apply the Pythagorean theorem to this right triangle.

[00:15:03]So let's go.

[00:15:04]Since I marked YB and Ya in blue, I know what this leg here is?

[00:15:16]O YB o Y a tasco ali, se eu projetar para eixo das ordinates.

[00:15:28]Good evening, Wellington. You're running late, but I'm not going to make a note on your card, you 're free to go. Completely shut down. The other leg, by analogy, is the leg that will provide the abscissa information, which is XB or XA.

[00:15:55]Ready. So what do I do now?

[00:15:57]Pythagoras in it. Pythagoras in it. What's going to happen? I'll even put it in color there.

[00:16:07]The square of the distance AB is the sum of the squares of the legs.

[00:16:17]Are you forgiven? This guy studies in another class. I'll take the difference Y between the ordinates and the difference between the abissae.

[00:16:32]XB - XA² + YB - YA².

[00:16:42]I actually prefer working with this formula here. I don't really like doing my roots because they never grow out right.

[00:16:56]My roots are growing out a bit crooked. Then I'll do the root, I do it like this, see. And I don't like writing root.

[00:17:05]If you choose not to write the root, what can you leave as an option? The square of a distance is equal to the difference of the abscissas squared plus the difference of the ordinates squared. Exactly like this formula here.

[00:17:28]OK? He demonstrated one formula, I'll demonstrate them all, there's no need.

[00:17:33]To help you understand where this distance between points came from. So you're not going to be drawing a pretty triangle, creating a Cartesian plane, no. You're already on autopilot, you're just going by the book.

[00:17:50]So here's the thing, everyone, pen and paper, pencil and paper, it doesn't matter what you have to write. Next, we're going to do two practice exercises.

[00:18:03]Letter A, I have points one and one.

[00:18:07]What does that mean? First point, X is 1, it's 1. And the second point, 3 and 3. What is the distance between these two points, teacher? I go there, draw the Cartesian plane, mark point 1, and then mark point 33, and do it using plane geometry. There's no need.

[00:18:38]Go straight to the formula. I'll remind you of the formula below. I 'll even put it in the same color.

[00:18:50]The distance squared.

[00:18:59]Difference between x-coordinates.

[00:19:03]Here, writing Y first or X first doesn't matter. It's the legs of the right triangle, there's absolutely no problem.

[00:19:14]Before anyone asks, what is this? It's electric. I did n't understand the question. We are in an analytical geometry class, part of the mathematics course for Petrobras. Let's solve part A.

[00:19:33]Here, we'll use the squared distance between the points.

[00:19:41]No, I didn't name the points, so you do n't need to. You just need to do the following. X from one minus x from another 3 - 1 a quad added.

[00:19:59]It started with x, now it needs y as well.

[00:20:02]3 - 1.

[00:20:11]What is distance squared?

[00:20:14]2² + 2² distance squared is what? 4 + 4 = 8.

[00:20:29]If I want the distance by extracting the maximum, right, that I can take from the radical, the square root will be 2√2.

[00:20:45]Therefore, the distance between these two points in letter A is 2√2. Now I want you to quickly calculate, or have already calculated, the distance between these two points in letter B.

[00:20:58]Same formula, same thing. Start setting it up, I'll do it with you guys in a bit.

[00:21:07]Just looking isn't enough, man, Cadu. But it's like I won't have to do it in fifth grade, sixth grade, Aunt Teteca. I ca n't just sit there watching the class. No, it's not law here, it's not other subjects where you just look at the material. You have to get your hands dirty to understand how this distance between points works.

[00:21:31]Pretty cool?

[00:21:34]Easy, guys. Very easy. The distance was determined by a simple application of Pythagoras' theorem to a right-angled triangle on the Cartesian plane. What happens to B?

[00:21:51]The distance squared. It doesn't matter whether you start with difference X or difference Y. Let's start with X.

[00:22:02]-3 -3 equals -6².

[00:22:11]Now let's move on to y. 4 - 2² 36 + 4.

[00:22:28]The distance squared is equal to 40.

[00:22:36]Taking the square root of which we can extract, 40 x 10, a√4 is 2. So our distance is 2, and inside the square root is 10. 2√1. Oh, that's ugly, professor. It doesn't matter. That's the answer. He's going to put it in, he's going to win the point, he's going to avoid the problem.

[00:23:02]Beautiful.

[00:23:08]Wonder. What might the examiners ask you? Six Gran Rio usually asks. We'll look at some of those issues later on.

[00:23:20]Given three points on the Cartesian plane, determine whether these points are collinear or non-collinear.

[00:23:37]If they are aligned, like it is here in the drawing, look.

[00:23:43]Oops.

[00:23:48]A, B, and C.

[00:24:00]In this case, in our drawing, the points are aligned. We also won't have much time to explain each formula individually because we're using this formula with the points aligned.

[00:24:16]But basically, for you to understand, it came from the following calculation.

[00:24:28]Plotting the coordinates of each point here and drawing these triangles here, [snoring] what do I notice?

[00:24:55]This angle alpha is the same as angle alpha. What is the tangent of alpha, everyone? We 'll look at this more calmly later on.

[00:25:06]The slope that causes the line to do this, that changes the angle of my line, is the tangent of alpha. It is the change in y divided by the change in x. So, between B and C and between A and B, the tangent of that alpha is the same, because the angle is the same. You equate them, you equate the slopes, because the line that passes through A and B is the same as the line that passes through B and C. So the slope is the same. By equating the coefficients, you arrive at this formula here. It falls into this formula here. So you've already learned the following. When the three points are in a row, when the three points are in a row, you just have to make the identity matrix there, the last, the last column, all one, first column, all X, second column, all Y. So, how do you do that?

[00:26:31]So, there you go.

[00:26:33]First row: X of A, Y of A, and 1. Second row of your matrix: X of B, Y of B, and 1. Third row: X of C, Y of C, and 1.

[00:26:56]And you solve this determinant and set it equal to zero.

[00:27:05]Either you solve it, if he's asking you if the points are collinear, you'll see if the determinant equals zero, they are, look here, determinant equals zero.

[00:27:22]They are lined up here. I'm making a small correction: here, the determinant is not zero. When the determinant is different from zero, then what do these points automatically form? Triangle.

[00:27:45]Because they are vertices of a triangle, they are misaligned.

[00:27:49]Pretty cool?

[00:27:52]So, you've already learned how to interpret given three points, whether they form a line or a triangle.

[00:28:05]All you have to do is take what? This matrix here, where the coordinates are enclosed in red. And find the determinant.

[00:28:20]The determinant will define whether it is a straight line or a triangle. Hey Cadu, how's it going?

[00:28:29]I thought that the determinant is different from zero.

[00:28:36]All good? It's a triangle. So those three points there, look, form a triangle, like the one shown in the figure. How do you find the area?

[00:28:46]The examiners also ask a lot about the area. This is the area right here.

[00:28:56]Half of the determinant.

[00:29:00]Half of the determinant. You have to record it, teacher. Yes. Record it there. The area of ​​the triangle.

[00:29:10]The determinant will give you a value. 20, 30, 40. If the result is 40, what is the area in square units? Remember that this area won't be measured in centimeters; it won't be measured in centimeters. It will usually come in area units.

[00:29:25]If you found the determinant to be 40, then your area is 1/2 of 40. 20. An example. Where will this determinant value come from? Cadu?

[00:29:43]If you, when you find the value of the determinant using Sarros' rule for the matrix, it's a done deal.

[00:29:53]So, what do we have to do? Practice exercise.

[00:29:59]I'll give you the points and we'll determine if this case forms a triangle or not, or if it's a straight line. Come with me. It helps me.

[00:30:13]Letter A.

[00:30:15]Points 1 2 3 6 5 10. It's there.

[00:30:22]So, how do I set this up?

[00:30:27]A B C. Let's put them here on the side.

[00:30:32]Tum tum tum tum.

[00:30:35]Let's go. Our.

[00:30:37]headquarters.

[00:30:41]First line X [snoring] from A. Oops, remembering. Let's go. Here, look.

[00:31:00]X of A, Y of A. Oops, let's see, let's see, let's see, let's see.

[00:31:07]It happened.

[00:31:13]So, uh, here we are.

[00:31:21]One, two, end of discussion.

[00:31:26]3 6 coordinates of B. And now of C.

[00:31:34]5 10. First line, the coordinates of A. Second line, the coordinates of B.

[00:31:41]Third line, the coordinates of C. No problem at all.

[00:31:47]1 1 1. Closed? Sarros' rule tells us the following: take the first two columns, replicate them on the right side of the matrix, and then multiply the main diagonals.

[00:32:03]subtracted from the secondary values.

[00:32:06]Here, let's add another color just to differentiate them. I'll take the 1 35 and put it here.

[00:32:15]I'll put the 2610 here.

[00:32:20]But you, Bruno, Robson, Samara, Won, who became an expert on matrices, what did you realize? If you've studied matrices extensively, you should know that matrices are also part of our course syllabus.

[00:32:36]When I have it, it's right here, I'll schedule it for you. What is a linear combination? Proportional queues. What are proportional queues?

[00:32:48]A queue can be a column or a row. Queue. Queue. They are both in line.

[00:32:57]If you look at this column and this column, which is the first and second column, the second column is double the size of the first, right?

[00:33:22]By property, when the queues are proportional, what happens? What's going on, Robson? A zero determinant is automatic.

[00:33:41]Professor, I want to do this determinant.

[00:33:45]You can do it, you can do it. How do you do it? I'm just going to remind you here, this time this time this result here. I'm going to delete this here to make some space.

[00:34:01]This times this times this.

[00:34:03]Positive result here. Let's do this real quick.

[00:34:07]+ 6 + 10 + 30 and what are the secondary functions? It's multiplication, but you put in the negative -6, -30.

[00:34:38]And then you realize what you're left with? Everything symmetrical. -30 is canceled out by 30, -10 is canceled out by 10, -6 is canceled out by 6, and zero. Just realizing that the lines are proportional is enough to solve the problem. So, what happens with the letter A?

[00:34:56]You answer like this, look.

[00:35:00]Letter A.

[00:35:05]Colinei Aráes. Collinear.

[00:35:13]Great. That's great, everyone. If you don't remember what "matrix" means, go back a little bit. How does the matrix weapon work?

[00:35:21]How do you apply the rule of thumb that is used for a 3-matrix? Three rows, three columns. Beauty? These properties help you quickly determine if the determinant is zero. And here, what matters to us is knowing if the determinant is zero.

[00:35:42]Let's look at letter B.

[00:35:45]Here was A. Do letter B over here.

[00:35:50]The letter B is as follows. First line 40.

[00:35:58]Second line 00 06.

[00:36:04]Coordinates of A, coordinates of B, coordinates of C. Prove who A, B, and C are.

[00:36:10]A is the first one there, B is the second, and C is the third. The third column completes everything with a one.

[00:36:20]Close. Now for the rule of sarcasm. The first two columns are replicated on the right.

[00:36:27]4 0 0 0 6.

[00:36:36]Oh, Cadu, there are a lot of zeros in that thing.

[00:36:40]This thing will give a zero. Careful, careful. Quickly apply the rule of mockery.

[00:36:49]Multiply everyone here.

[00:36:52]Zero.

[00:36:54]Zero here as well.

[00:37:00]Zero here as well. Now, secondary diagonals.

[00:37:08]zero. Oops, here it came out -24 and here it came out zero. This determinant gives -24.

[00:37:28]So, what do you do?

[00:37:31]Answer down below.

[00:37:37]It forms a triangle because the determinant is not zero. The determinant is different from zero. It's -24.

[00:37:56]And now, quickly, you who figured out the area formula, okay, professor.

[00:38:03]A, the letter B will give me a triangle as the shape. What is the area of this triangle?

[00:38:15]1/2 of the determinant.

[00:38:19]Units of area. Area cannot be negative, that's why we use absolute value, OK, everyone? Half of 24, because half of -24 is -1. What is a module like? The absolute value is always positive. Area, a unit of area, is always positive. If you wanted to know the area of ​​that triangle in letter B, 12 is area units, you would find that little U next to the number, which is the area unit. Completely shut down.

[00:38:57]Wonderful. So what do we have now?

[00:39:07]our equation of the line. I'm just going to reinforce the points here because it's a little faded.

[00:39:17]A, not B, here it's APB.

[00:39:23]A P B.

[00:39:29]OK.

[00:39:31]Here's the thing, remember this very well, you're going to have to memorize this formula, folks, where A is the coefficient of X, B is the coefficient of Y, and C is the independent term. What is an independent term? There is an independent variable that has no coefficient; nothing is multiplying it. This is the formula for the general equation of a straight line. And what does this equation lead to?

[00:40:15]That one, just so you understand, comes from the determinant obtained when the points are collinear, okay?

[00:40:27]Everyone lined up perfectly. When everyone lines up perfectly, they form a straight line.

[00:40:32]Ah, so by solving this Cador determinant, I will obtain the general equation of the line.

[00:40:41]Yes.

[00:40:42]So you memorize the equation of the line and remember that it came from the concept of collinear points. Aligned points form a straight line.

[00:40:54]If you want to remember who Q is, look, A, B, and C are coefficients.

[00:41:06]Coefficients.

[00:41:12]This C here, let's even put another one, another color, is called the independent term.

[00:41:28]Wonderful.

[00:41:32]How can that be asked, professor? Given the coordinates, let's do practice exercise three together.

[00:41:40]Given the coordinates, find the line that passes through two specific points.

[00:41:47]So, how do I do it? You can do that by aligning points with our matrix, our determinant. You can kill that in no time. Also remember that the concepts of first-degree equation and first-degree function can all be applied here, because you must have thought, if you are good at math, what represents the first-degree equation, the first-degree function, a straight line. Ah, so I can use all the concepts of linear functions. Or you can simply take the alignment matrix, which is much easier, and plot those points. Then you find the equation of the line.

[00:42:46]Didn't see it? Come with me here for a second.

[00:42:54]Here we'll put the letter A so you know that this one is for the letter A. Up there you put XY. Why?

[00:43:07]Because it's as if you don't know the coordinates of the first point. You put XY. Then, on the second and third lines, you put the coordinates of the points you already have, which are already given here. 1 and 2, 3 and 6. 1 closes.

[00:43:37]Sarros rule.

[00:43:40]He replicated the first two.

[00:43:47]Beautiful.

[00:43:50]Just a reminder, close up and let's go!

[00:44:05]Main diagonals.

[00:44:10]2x here + 3y here + 6. Where is this result coming from, Cadu? You multiply everyone who's on that diagonal. 1 x 1 x 6 equals +6. Main diagonals, positive results. Secondary diagonals result in the opposite sign.

[00:44:46]Here - y, here - x, and here - 6. That 's correct. Pam p. Okay.

[00:45:08]Can I cut something?

[00:45:12]This one with this one? Of course, of course, right?

[00:45:17]They are aligned. You forgot, Cado, to indicate there that if they are a straight line and we are finding a straight line, it is necessary that this determinant be zero. So we put it here, yes, of course, just a straight line. Now we just arrange the X with X, the Y with Y to make it more organized.

[00:45:50]- 4x means that I killed this one with this one, taking y + 2y.

[00:46:06]I killed this one with this one, and it ended up equal to zero here.

[00:46:14]Equal to zero, professor. But this equation will be left without the 'c', without that independent term.

[00:46:21]It can happen, there's absolutely no problem with it.

[00:46:24]This is the format of the general equation, always starting with X, then Y. This one doesn't have any independent terms. So here you've provided the equation that passes through those points mentioned in letter A.

[00:46:53]Therefore, any coordinate you put in there will work because of the proof that it will work.

[00:47:06]in this equation. Come on, otherwise everything's wrong. Otherwise, we're crazy, we're nuts. If I put here, see, -4 multiplied by 1 + 2 multiplied by y, 2 has to equal 0, so -4 + 4 equals 0. Do you see? What does that equation represent? The alignment of those two points. The straight line that passes through those two points perfectly.

[00:47:37]Beauty? Help me do B. It's obsessive.

[00:47:41]You have to do it too. I'm going to do it right here in this little corner.

[00:47:49]He made the mold, he set it up. What do you put on the first line?

[00:47:57]X and Y. In the second, the coordinates of the first point, then the coordinates of the second point. The last column is one, one. Close it, replicate the first two columns, close it, and then what? equal to 0.

[00:48:25]Because they are aligned, it's a straight line you're dealing with, it has to be equal to zero.

[00:48:31]Quickly, let's see.

[00:48:37]5x positive 4y positive + 18. Now let's look at the secondary diagonals, going down to the left. - 2y - 9x 20 - 20.

[00:49:15]It was wonderful. Now let's see what can be cut and what can't be cut.

[00:49:26]Or maybe we should just start with the x.

[00:49:30]I have - x.

[00:49:36]Where do I have more x's? Look, 5.

[00:49:39]So, what's the result of this?

[00:49:46]- 4x. So I killed that one with that one. Now you go over there to the y. Anyone who has y - 2y + 4y gets + 2y.

[00:50:03]So you killed that guy with that guy. The only thing that remained was the term " independent." - 20 with 18 - 2 = 0. Just remembering that here it equals zero.

[00:50:21]Wonder. It is in the format of the general equation.

[00:50:32]Beautiful.

[00:50:35]Ah, Cadu, everyone is a multiple of two. So, would writing this equation or a simplified equation divided by two be the same thing?

[00:50:49]Yes, you can simplify a number, a fraction, or you can also simplify an equation directly. What would remain? Everyone, divided by 2 - 2x + y - 1 = 0, would be more concise.

[00:51:10]More streamlined.

[00:51:12]Beauty? And I'm going to teach you the simplified equation right here. This is the general equation.

[00:51:21]The simplified equation is when you isolate y.

[00:51:28]Move the y-coordinate to the left and everything else to the right on the right side of the equation.

[00:51:37]2x to the other side.

[00:51:42]So it's like this, see?

[00:51:45]This here is called, let's put it here, the reduced equation.

[00:52:00]She'll be a little further ahead, but I'm giving you a heads-up now. It's the same equation, the same line with the same properties. Only the way of writing is different. It's extremely important.

[00:52:17]Slope.

[00:52:19]What is the slope?

[00:52:22]What I really want you to record is this right here.

[00:52:26]It's the tangent of alpha. Why is it the tangent of alpha?

[00:52:33]Going back to the first slides, alpha is this angle here.

[00:52:45]Where?

[00:52:48]Alfa is this one right here, folks. I'll write it here.

[00:52:55]That angle that's shown in red on the first slide. So, if I take the tangent, remembering that the tangent is what?

[00:53:08]the opposite side over the adjacent side. Therefore, when I say later on, the tangent of alpha is the difference between the ordinates YB - YA and XB. So it's the opposite leg. This guy here is the opposite leg.

[00:53:36]This guy here is the adjacent side. And that's what happens there, look.

[00:53:46]The tangent of that alpha is exactly like that quotient, right? This is the division between the difference of the ordinates Y over the difference of the abscissas. If you want to bring it into that format, and from the general equation, it is -a over b. This tangent, right, which can also be called - a over b. 'a' is the coefficient of x, the coefficient of y, and A is the coefficient of x in the general equation, in the form of a general equation. The coefficient of Y, B, in the form of a general equation.

[00:54:42]Wonder. Everything alright, Cadu?

[00:54:46]Remember that the simplified equation has been brought here again, y equals MXB.

[00:54:58]This M will appear a few times.

[00:55:02]Don't forget that in the reduced equation, in the reduced equation, that m there, the m, folks, is what makes the equation do this here, see.

[00:55:16]U.

[00:55:18]So this M here is [snoring] the slope.

[00:55:30]What is slope? That changes the angle. What is the linear coefficient? It's because it affects the position of the line.

[00:55:39]So B is the linear coefficient. Basically, you have a straight line here. What does M do? That's with the straight line.

[00:55:56]And B does this here with the straight line, look.

[00:56:01]She's constantly shifting from one position to another.

[00:56:08]Wonder.

[00:56:09]And this formula here is the formula we use when faced with the reduced equation, given what?

[00:56:21]A coefficient and a known point.

[00:56:27]He'll tell you like this: The line that passes through a point, you see those two little dots there, right? The straight line that passes through a given point and has a known slope.

[00:56:40]Slope is the angular coefficient.

[00:56:44]You use this formula, take the slope given in the problem, the point given in the problem, and you can quickly find the line.

[00:57:00]Oh, Cadu, help me, help me. There's going to be an exercise where we'll do this and we'll ace it. What falls most often?

[00:57:13]Last topic. The most important issues were discussed here. Take a breath, generate some energy, save a little ATP so you can understand these relative positions between two lines. This is also covered in the systems section.

[00:57:31]How to discuss a system and how to analyze a system, whether it is possible and determined, whether it is possible and indeterminate, or whether the system is impossible. Why are you analyzing this? Based on the position of the lines, because each line is an equation, what is the system of equations?

[00:57:56]Uh, uh, solving several equations together, a solution that seeks what? The solution seeks a pair that works for both equations.

[00:58:12]So let's go. Let's look at the lines; I'll write R and S on all of them so you understand. This is R.

[00:58:27]This is S here. This is R.

[00:58:33]This is S. Here. This is R.

[00:58:37]Here. This is S. In the first drawing, what happens?

[00:58:42]The lines are parallel.

[00:58:45]According to the drawing, competitors. Competitors are crossing each other, and in the third drawing, they are perpendicular.

[00:58:57]So, what does each of these positions generate?

[00:59:01]a relationship between the slopes. Let's start with the first one here.

[00:59:17]What does the M in R mean when it's different from the M in S, Cadu? If the slope of one is different from the slope of the other, they automatically intersect.

[00:59:28]So they are competitors.

[00:59:32]So you don't need to draw the lines, you don't need to draw.

[00:59:35]If you have the standard equation of the line, you just need to analyze the slope.

[00:59:44]M is different from intersecting lines.

[00:59:49]What does that mean? Remember, if he asks you about systems of equations, if you use these equations to solve the system, what do you get?

[01:00:03]A possible and determined system.

[01:00:08]Why? There's a point there that's the solution to your system. Hey Cadu, how's it going? Second, the second line, when the slopes are equal, well, but B, which is the y-intercept, is different. So this is the first drawing.

[01:00:33]They are parallel, but distinct.

[01:00:38]Parallel, but distinct. Why are they different? Because this B, which is the linear one, is different from the B of one line compared to the B of another line.

[01:00:54]Their linear coefficients are different. What does that suggest? Parallel, but distinct. So what do we have here?

[01:01:03]A possible system, but an indeterminate one.

[01:01:10]And when we have equal slopes, but also equal y-intercepts, then it's a flat line, there's no drawing like that, but what are they straight lines? Straight lines. Suppose that in this drawing, the first drawing there, R and S meet, stick together. So they are coincidental. If they coincide, the system is impossible; it has no solution. And the last one, the very last one, is your third drawing. When the slope of one is the negative reciprocal of the slope of the other, they are perpendicular.

[01:02:00]He said they are perpendicular. What does perpendicular mean?

[01:02:03]Orthogonal stairs.

[01:02:05]It forms a 90º angle.

[01:02:08]So, is it possible to relate these slopes? From the. The coefficient of one line is the negative reciprocal of the other. It's right there.

[01:02:19]Let's do an exercise to try and solidify this.

[01:02:22]Try not to. If you understood the concepts discussed so far, you'll ace these exercises. It's nothing to be afraid of.

[01:02:36]You will try and you will succeed.

[01:02:38]Let's go.

[01:02:41]Number one. Consider R the line represented by the equation 2y - x - 1 = 0. And the point P1 given by the ordered pair XY such that X = 2, Y = 4. Both are in the XY plane.

[01:03:00]Let R2 be the perpendicular line. Look at this important information.

[01:03:08]Perpendicular to R1. He said that R2 is perpendicular to R1. Passing through point P1, he wants to go to point P2. intersection between lines R1 and R2. He wants the ordered pair that represents what?

[01:03:24]P2 and P1.

[01:03:31]So let's go.

[01:03:33]Look at this, everyone, how interesting! There is a given point, which is point 24, point P1.

[01:03:44]He's given 2 4 and what does he say?

[01:03:55]And it goes in a straight line. Ah, he gives a straight line and says the following, look.

[01:04:04]R2 is perpendicular to R1.

[01:04:11]So here's the thing, we can find the coefficient of R1 because the line for R1 is given, it's right there. R1 has a line represented by this equation. What is the coefficient? If you remember some of those slides, - a over b, it's here, here - a over b. So we put the coefficient of the line R1 here. What?

[01:04:54]Less.

[01:04:56]Careful, very careful, because here he put the letter B in front.

[01:05:03]B is the one who accompanies Y. A is the one who accompanies X. He put it there on purpose so you would get it wrong.

[01:05:10]Pretty cool? So, look, A minus -1 will give you plus 1, right?

[01:05:24]Show.

[01:05:27]Which is less - a about B.

[01:05:31]Hmm.

[01:05:33]So, automatically, what is the one in R2? The negative inverse.

[01:05:42]What is the negative inverse of that?

[01:05:46]Why?

[01:05:48]Because these two are perpendicular.

[01:05:56]Using the formula, you take the slope here and the negative reciprocal of the other coefficient. Hey, Cadu. So I take this coefficient and the given point, which is 24, and what am I going to find?

[01:06:19]The line R2. I already have the R1 line. All I need to do is find the other straight line. How am I going to find the other line? Let's go. Here, look, the formula, look. What is the formula? This one here. Y - Y is equal to the coefficient that multiplies X - 1 by any x.

[01:06:45]So, let's go.

[01:06:47]What is the given value of y? A given point is a datum.

[01:06:51]Given point 2 4. So here we put 4. The coefficient. We're finding this one here. Let's put the equation here: R². So let's take the coefficient found for R2 -2.

[01:07:24]Given a point with coordinates 2x2, then y = 4 - 2x + 4.

[01:07:41]Amazing, amazing, amazing.

[01:07:44]What are we going to do? Since the first equation, the first line, is in general form, let's put everything else into general form. We're going to put this one in the general format. Stay like this, see?

[01:08:01]2x moving to the left + y - 8 = 0.

[01:08:14]Now let's take the other equation, which is R1, it's up there.

[01:08:24]-X + 2Y - 1 = 0. And now I solve this system, because by solving this system between lines R1 and R2, I will find the ordered pair, which is the solution. I'm going to find the ordered pair, which is the solution. So let's solve this little system quickly. Why?

[01:08:56]because of the fastest format, which is addition and subtraction.

[01:09:01]First, I have to change my clothes a little so I can add or subtract this correctly.

[01:09:09]Let's go. If I multiply this second value by two, what does it become?

[01:09:17]Let's put it down here.

[01:09:23]Things will remain the same on top.

[01:09:28]Below it will be -2y + 4y - 2. Now what happens? If I add the two together, what will disappear? The 2x will become 5y. I am adding y to 4y. 5y - 10 = 0.

[01:10:07]5y = 10. Moving to the other side, we find our y = 2.

[01:10:17]Oops.

[01:10:19]Only then. Look, I'm telling you that Gran Rio isn't as difficult as it seems.

[01:10:26]Just finding the Y and you're good to go. Letter C. You don't even need to find the X.

[01:10:36]No, Cadu. I want to kill the snake, show the stick, call the forensics team, take fingerprints, right? To collect, send to jail, the stick, everything's in order.

[01:10:50]Replace it here.

[01:10:56]Replace it here. 2x + 2 - 8, right?

[01:11:09]2x - 6 = 0. 2x = 6 x = 3. He found x and ran to the crowd anyway. What is the ordered pair?

[01:11:24]X is 3, y is 2. I used everything the problem gave me. Everything that the problem gave me.

[01:11:33]One more.

[01:11:35]What is the reduced equation of the line that contains the height, oops, of the side relative to side BC of triangle ABC, where A, BC are the points 3 4 1 6?

[01:11:53]So, let's go.

[01:11:57]He wants the height relative to side BC.

[01:12:12]Can I find the slope of BC? With you.

[01:12:21]Let's start with him.

[01:12:29]The slope of BC is as follows. I take the difference between the y's and divide it by the difference between the x's. So, let's go.

[01:12:40]I'm just going to list here who is B, who is C, and who is A so you know, see?

[01:12:47]A, B, C. He explained this in the problem statement. Awesome!

[01:12:54]So, Y = BC, Y = C is 0, -1, and 6 = -1.

[01:13:09]Here it became -1/5 -1/5. So what's going on, folks?

[01:13:24]So what did I find? The angle of side BC. If I want to find the slope of the height, well, the height is this: For example, if side BC is here, the height is measured here. So, this line, which is the height, has a slope-to-angle relationship. What?

[01:13:59]negative inverse.

[01:14:02]Inverse negative.

[01:14:04]So let's put it here, the coefficient of the height BC is the negative reciprocal.

[01:14:29]It's C. Hmm, interesting.

[01:14:33]So, let's do this, everyone. You have to realize that the relative height comes from point A.

[01:14:46]So it passes through point A and has the slope of the height, which we already found to be five. So we use the equation where we have the slope and we have a point.

[01:15:03]What is the equation?

[01:15:06]In this case, Y minus Ya, the slope is 1 X o XA. So, let's go.

[01:15:20]What is Y? It's the one from point A, because it will pass through point A, the height, and it will make a 90º angle with that point on side BC.

[01:15:31]Four. The coefficient, which we've already found, is the slope of the height 5X minus the X of A, which is 3.

[01:15:46]Perfect, 5 multiplying what's inside the parentheses.

[01:15:53]5x - 15. He put the alternatives in reduced form, reduced equation format. So what do I do?

[01:16:06]I'll leave the x there and move that -4 to the other side. What remains? + 4.

[01:16:16]Done. Awesome!

[01:16:20]Here it is. And we're left with option A. Y = 5x - 11.

[01:16:33]One more, one more to fix three.

[01:16:42]Keep in mind that they all came from the six different Gran Rio races in different years. Pretty cool?

[01:16:51]The line T of the equation X + 3Y + K shown above intersects the coordinate axes at points P and Q. It's there.

[01:17:01]Point Q.

[01:17:05]OK. It is known that the zero point or O is the origin of the Cartesian plane, and that triangle POQ has a side length of 24 units.

[01:17:15]Important daily unit. What is the value of the real constant? K.

[01:17:24]Look, what connection can I make?

[01:17:31]Given a line with a slope, if I have the line, I have A and I have B, what can I calculate? What is this coefficient here, of line T? - A about B.

[01:18:00]Maravela. I'll retrieve the data from the equation.

[01:18:05]Who follows X? 1. So here it becomes -1 and B is 3. Oops.

[01:18:18]I found here - 1/3, which is the slope.

[01:18:26]Wonder.

[01:18:27]But look, look how interesting this point is here, taking the triangle, this triangle here, what happens, folks?

[01:18:54]The straight line. That's a first- degree function. Where the line intersects the axes is where one coordinate is zero and the other coordinate is zero. If I take this triangle here, what is it?

[01:19:12]This y-coordinate intersects with this x-coordinate, which is where the line intersects the axes. What is she?

[01:19:26]Taking the area of ​​remembering as a reference, let's remember, let's remember. Area of ​​the triangle: base height divided by 2. So, he said the area is 24.

[01:19:46]Therefore, this area of ​​24 has x as its base.

[01:19:52]The height is y over 2. So my x y, what is it?

[01:20:01]48.

[01:20:05]So I know that x y is 48, but I have this alpha here, look.

[01:20:12]Remember, alpha, alpha is this one here.

[01:20:18]But I want to relate this one here.

[01:20:29]So, relate the tangents in the following way. What is the tangent of supplementary angles? I 'll just remind you. Let me do this quickly.

[01:20:47]True, sine, cosine.

[01:20:55]This here is the tangent axis.

[01:20:59]I'm drawing any angle alpha here. What does this alpha angle represent?

[01:21:17]tangent, this part here on the trigonometric circle. Just so you understand, we're going to relate the tangents.

[01:21:28]If I replicate this alpha here, what do I want to know now? What is the tangent of this angle here?

[01:21:39]Tangent of this angle here, look.

[01:21:51]All I have to do is replicate it here to reach the tangent axis, and I realize it's exactly the same tangent, only negative. So, supplementary angles, that's all to remind you of them here. In supplementary angles, the tangent of one is less the tangent of the other.

[01:22:22]Hmm.

[01:22:25]Cool? So, Cadu, this tangent here of this angle, for example, this -1/3 here is relative to this angle alpha, this slope of -1/3 is relative to the angle alpha. But if I relate the tangent that's in there, which is 180 - alpha, to be able to relate yx, it becomes, it becomes plus 1/3. So, what am I going to write here?

[01:23:13]A little system like this, look.

[01:23:17]Y - x, why? tangent, oh, opposite side over adjacent side. The tangent of this guy is less the tangent of that other guy. The tangent of 180 - ala is minus the tangent of alpha.

[01:23:36]But the tangent of alpha is -1/3. So I write 1/3.

[01:23:42]I know that y so x is 1/3. And I know that XY is 48.

[01:23:53]That's it.

[01:23:55]Now I've nailed it.

[01:24:00]Why? Here's a system that can be solved quickly. x = 3y.

[01:24:16]I substitute x = 3y down there.

[01:24:20]X that multiplies. Oops.

[01:24:24]X that multiplies no.

[01:24:27]If x is 3y, it will be 3y multiplied by y, 48.

[01:24:44]y² 16 y⁴ or y - 4.

[01:24:56]Why don't I use -4?

[01:25:01]Based on the design. From the drawing there, you can see that the 'y' we found is what? The positive side.

[01:25:11]Legal.

[01:25:16]What is x? Just replace it here.

[01:25:21]X = 3y. X is 12.

[01:25:26]Then I take this equation here and find K. X is 12 + 3Y 3 x 4 12.

[01:25:55]Whoa, no, hold on.

[01:26:00]I've already found Y, I don't even need to find X.

[01:26:02]I'm crazy, crazy, crazy. Okay, folks. This point here, look, this point that passes through the line, remember, look.

[01:26:18]I'm using this point here, which is 40.

[01:26:27]This system I used is to find the X and Y coordinates where the line intersects the axes. Just so you understand, look. So, whoa, um.

[01:27:01]Now, just here for [sighing] now. I don't know if I did the right thing here.

[01:27:18]Finally, to wrap things up, we're asking for help from university students. I accidentally switched the presenter mode here. Hold on a minute. Let's go back.

[01:27:29]Leave it as it is. Just come back.

[01:27:34]Just come back here.

[01:27:38]Will we come back like this?

[01:27:41]That's the format, this is the format.

[01:27:47]Everything's resolved.

[01:27:49]Intelligent people are something else entirely, my friend.

[01:27:52][laughter] Let's go. Oh no, he ran out of notes. I don't believe.

[01:28:24]Okay, folks, I forgot the notes here, but we'll figure something out. What was this place? It was point 30 and here is point 120. So what are we going to do on this line here?

[01:28:45]We're going to set x to 12 and y to 0.

[01:28:59]And we'll find K, which will be equal to -1.

[01:29:13]- 12.

[01:29:16]Closed off. Beautiful. Here's letter A.

[01:29:22]Let's speed up the others here because this is a little faster. The vertices of triangle ABC are the points 0, 2, 4, 6, 8, and -10. The XY coordinates of the midpoint of the longest side of triangle ABC are the midpoint. Let's go. Look, from the larger side. So, we're going to use the distance formula from all sides. First, we do the following: AC, BC. I'll take the one from AB, I'll take the one from AC, and I'll take the one from BC. AB.

[01:30:15]AB 4 - 0² 4 6 - 2 4² Great.

[01:30:27]It's AC 8 - 08² and - 10, I have AC - 10 12. Oops, 12 qu professor.

[01:30:44]But it's - 12 qu, it doesn't matter. When squared, the result will be positive. Have you noticed that the AC is already bigger? I don't even need to know that AC is larger than AB. And what about BC?

[01:30:58]BC 8 - 4² - 10 menã 16.

[01:31:12]Quad.

[01:31:13]Wonder.

[01:31:17]Forget about this one.

[01:31:18][sighing] But there, AC and BC, we'll have to calculate to know which is the larger one.

[01:31:24]So here's what's going to happen.

[01:31:28]64 plus 1 = 44. So what will go here?

[01:31:35]16 plus 256, performing the necessary sums. 208 versus 272. Professor, but you don't even need to extract the square root to know that. By squaring the distance AC and the distance BC, you already know which is larger. Here it is, BC. Midpoint. The midpoint is calculated by adding the coordinates. by two.

[01:32:14]BC 8 + 4 12 6 BC coordinate Y 6 with -10 - 4 - 2. There you go, that's what you wanted. That's faster. Coordinates 6 and -2.

[01:32:34]Letter C.

[01:32:37]Last, last.

[01:32:40]A rectangle ABCD is drawn on the Cartesian plane. I'm drawing this here just for us, I'm not just going to remind us, look.

[01:32:58]Given vertices A, C, and D on sides A, B, and CD, find the equation of the line that passes through B and the midpoint of side CD. So, basically, that's it, right?

[01:33:17]Wonderful.

[01:33:18]So, what's interesting about having what?

[01:33:26]I deleted too many.

[01:33:34]alpha. So, what we want is this: The equation of the line that passes through point B, through the midpoint of CD. We can even put any M here.

[01:33:47]So, all we need to know is what?

[01:33:52]The slope.

[01:33:58]Slope.

[01:34:01]It passes through point CD. What is the midpoint of CD? This M here allows us to quickly determine the average CD. All I have to do is take the coordinates CD, add X to X and multiply by 2, YY by 2.

[01:34:23]So there you have it, 8 plus 2, 10.

[01:34:28]And 25 plus 1.

[01:34:34]Beautiful.

[01:34:37]Let's go, everyone. This distance from here, remembering that this one here is the same as this one here, right?

[01:34:50]This guy here, C, has coordinates 8 1.

[01:35:00]So, all I need to do is find the distance AD ​​and replicate it for this distance BC.

[01:35:09]Oh, this distance BC is the same as AD. So, by finding this distance AD, I will find the distance BC.

[01:35:29]The distance from AD is as follows.

[01:35:32]A with A 0² + A with D, huh?

[01:35:38]A with D 5 - 1 4².

[01:35:45]The distance AD ​​is 16².

[01:35:50]No, sorry. 16.

[01:35:56]Here it is.

[01:35:58]This is the distance, folks. It's 4 here and 4 here.

[01:36:07]So what is the distance MC?

[01:36:17]What is the distance MC? Why would I need to find that MC distance? To calculate the tangent, I have this value. All I need to do is find this value here. I'm going to calculate the square of the distance from M.

[01:36:31]C has the coordinates, it's easy. It's 8 - 5 3² plus 1 - 1 0. So this distance MC 3² minus 3 is here.

[01:36:50]So this is 3.

[01:36:53]So what is the tangent of alpha that I want? Opposite side over adjacent side, 4/3.

[01:37:04]Just do that little equation that has the slope and the points, which is this equation here, see.

[01:37:13]Y - y slope X - o XA.

[01:37:21]Wonderful. The thing I want here, folks, is... Go past this point here, look.

[01:37:40]So, it's passing through point M and has a tangent slope of alpha.

[01:37:48]Point M has Y coordinates 1 and X coordinates 5.

[01:37:59]Coefficient 4/3.

[01:38:05]Wonderful.

[01:38:07]If I multiply this by three, why? To get rid of this 3 here at the bottom, it will become 3y - 3 4x - 20.

[01:38:32]Making this little corner here because the thing got shorter.

[01:38:38]The equation will remain as is, let's put it in the format that's in the answer choices, which is the general equation format. Bring the x, it's already positive. Let's move the 'y' over there.

[01:38:54]4x.

[01:38:56]This 3y goes to the right, becoming -3y.

[01:39:01]The -20 is already on the right. This -3 becomes +3, resulting in -17 = 0. Ah, it's there, it's in option B, look.

[01:39:21]4x - 3y - 17.

[01:39:28]Whew, phew. Phew, phew! Phew, phew. If you don't know planar physics, if you don't know spatial physics very well, maybe you don't know analytical physics either. Everything is interconnected.

[01:39:42]OK. Hey professor, I'm having trouble developing this. Go back a little to the plane, go back to the concepts of triangles, remember a little about first-degree functions, how to deal with the Cartesian plane, and then things will go much easier. It's not simple, you have to burn brainpower, gray matter, and ATP, but that's what it's for in life. If you want to be successful, you have to do something extra. Here you've got your butt in the chair, lots of motivation, but also a lot of sweat.

[01:40:14]We're closed. with our classic phrase by Teodor Roosville.

[01:40:20]Before that, hey, follow me there, guys.

[01:40:22]@matemática on Instagram, mathematics with Kaadu Cádu Araújo. It is far better to risk great things, to achieve triumphs and glories, even exposing oneself to defeat, than to line up with the poor in spirit, who neither suffer much nor enjoy much, because they live in that gray twilight and know neither victory nor defeat.

[01:40:45]I repeat that phrase throughout the entire class so it sticks in your mind. I thank everyone who stayed, Robson, Wellington, Samara, Bruno.

[01:40:58]Review it. If you have any questions, suggestions, comments, or messages, feel free to send them on Instagram; I'll be ready to receive them.

[01:41:12]Study hard, because the successful candidate is the one who never gave up. Good evening, hugs.