Coordinate plane (6th grade) - Knowledge Hypermarket. Start in science
Rectangular coordinate system on a plane
A rectangular coordinate system on a plane is formed by two mutually perpendicular coordinate axes X’X and Y’Y. The coordinate axes intersect at point O, which is called the origin, a positive direction is selected on each axis. The positive direction of the axes (in a right-handed coordinate system) is chosen so that when the X'X axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the Y'Y axis. The four angles (I, II, III, IV) formed by the coordinate axes X'X and Y'Y are called coordinate angles (see Fig. 1).
The position of point A on the plane is determined by two coordinates x and y. The x coordinate is equal to the length of the segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement. Segments OB and OC are defined by lines drawn from point A parallel to the Y’Y and X’X axes, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A. It is written as follows: A(x, y).
If point A lies in coordinate angle I, then point A has a positive abscissa and ordinate. If point A lies in coordinate angle II, then point A has a negative abscissa and a positive ordinate. If point A lies in coordinate angle III, then point A has a negative abscissa and ordinate. If point A lies in coordinate angle IV, then point A has a positive abscissa and a negative ordinate.
Rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY and OZ. The coordinate axes intersect at point O, which is called the origin, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units of measurement are the same for all axes. OX - abscissa axis, OY - ordinate axis, OZ - applicate axis. The positive direction of the axes is chosen so that when the OX axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the OY axis, if this rotation is observed from the positive direction of the OZ axis. Such a coordinate system is called right-handed. If thumb right hand take the X direction as the X direction, the index one as the Y direction, and the middle one as the Z direction, then a right-handed coordinate system is formed. Similar fingers of the left hand form the left coordinate system. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (see Fig. 2).
The position of point A in space is determined by three coordinates x, y and z. The x coordinate is equal to the length of the segment OB, the y coordinate is the length of the segment OC, the z coordinate is the length of the segment OD in the selected units of measurement. The segments OB, OC and OD are defined by planes drawn from point A parallel to the planes YOZ, XOZ and XOY, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A, the z coordinate is called the applicate of point A. It is written as follows: A(a, b, c).
Orty
A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.
In the three-dimensional case, such unit vectors are usually denoted i j k or e x e y e z. In this case, in the case of a right-handed coordinate system, the following formulas with the vector product of vectors are valid:
- [i j]=k ;
- [j k]=i ;
- [k i]=j .
Story
The rectangular coordinate system was first introduced by Rene Descartes in his work “Discourse on Method” in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method of describing geometric objects laid the foundation analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane.
The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.
see also
Links
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See what “Coordinate plane” is in other dictionaries:
cutting plane- (Pn) Coordinate plane tangent to cutting edge at the point in question and perpendicular to the main plane. [...
In topography, a network of imaginary lines encircling Earth in the latitudinal and meridional directions, with which you can accurately determine the position of any point on earth's surface. Latitudes are measured from the equator - the great circle... ... Geographical encyclopedia
In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with the help of which you can accurately determine the position of any point on the earth's surface. Latitudes are measured from the equator of the great circle,... ... Collier's Encyclopedia
This term has other meanings, see Phase diagram. Phase plane is a coordinate plane in which any two variables (phase coordinates) are plotted along the coordinate axes, which uniquely determine the state of the system... ... Wikipedia
main cutting plane- (Pτ) Coordinate plane perpendicular to the intersection of the main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide
instrumental main cutting plane- (Pτi) Coordinate plane perpendicular to the line of intersection of the instrumental main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide
tool cutting plane- (Pni) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the instrumental main plane. [GOST 25762 83] Subjects of cutting processing General terms of coordinate plane systems and... ... Technical Translator's Guide
Definition 1. Number axis ( number line, coordinate line) Ox is the straight line on which point O is selected origin (origin of coordinates)(Fig.1), direction
O → x
listed as positive direction and a segment is marked, the length of which is taken to be unit of length.
Definition 2. A segment whose length is taken as a unit of length is called scale.
Each point on the number axis has a coordinate that is a real number. The coordinate of point O is zero. The coordinate of an arbitrary point A lying on the ray Ox is equal to the length of the segment OA. The coordinate of an arbitrary point A of the numerical axis that does not lie on the ray Ox is negative, and in absolute value is equal to the length of the segment OA.
Definition 3. Rectangular Cartesian coordinate system Oxy on the plane call two mutually perpendicular numerical axes Ox and Oy with the same scale And common beginning countdown at point O, and such that the rotation from ray Ox at an angle of 90° to ray Oy is carried out in the direction counterclockwise(Fig. 2).
Note. The rectangular Cartesian coordinate system Oxy, shown in Figure 2, is called right coordinate system, Unlike left coordinate systems, in which the rotation of the beam Ox at an angle of 90° to the beam Oy is carried out in a clockwise direction. In this guide we we consider only right-handed coordinate systems, without specifically specifying it.
If we introduce some system of rectangular Cartesian coordinates Oxy on the plane, then each point of the plane will acquire two coordinates – abscissa And ordinate, which are calculated as follows. Let A be an arbitrary point on the plane. Let us drop perpendiculars from point A A.A. 1 and A.A. 2 to straight lines Ox and Oy, respectively (Fig. 3).
Definition 4. The abscissa of point A is the coordinate of the point A 1 on the number axis Ox, the ordinate of point A is the coordinate of the point A 2 on the number axis Oy.
Designation Coordinates (abscissa and ordinate) of the point A in the rectangular Cartesian coordinate system Oxy (Fig. 4) is usually denoted A(x;y) or A = (x; y).
Note. Point O, called origin, has coordinates O(0 ; 0) .
Definition 5. In the rectangular Cartesian coordinate system Oxy, the numerical axis Ox is called the abscissa axis, and the numerical axis Oy is called the ordinate axis (Fig. 5).
Definition 6. Each rectangular Cartesian coordinate system divides the plane into 4 quarters (quadrants), the numbering of which is shown in Figure 5.
Definition 7. The plane on which a rectangular Cartesian coordinate system is given is called coordinate plane.
Note. The abscissa axis is specified on the coordinate plane by the equation y= 0, the ordinate axis is given on the coordinate plane by the equation x = 0.
Statement 1. Distance between two points coordinate plane
A 1 (x 1 ;y 1) And A 2 (x 2 ;y 2)
calculated according to the formula
Proof . Consider Figure 6.
| |A 1 A 2 | 2 = = (x 2 -x 1) 2 + (y 2 -y 1) 2 . | (1) |
Hence,
Q.E.D.
Equation of a circle on the coordinate plane
Let us consider on the coordinate plane Oxy (Fig. 7) a circle of radius R with center at the point A 0 (x 0 ;y 0) .
Instructions
Construct three coordinate planes to have the origin at point O. In the drawing, the projection planes are in the form of three axes - oh, oy and oz, with the oz axis directed upwards and the oy axis to the right. To construct the last ox axis, divide the angle between the oy and oz axes in half (if you are drawing on a checkered sheet of paper, just draw this axis).
Please note that if the coordinates of point A are written as three in brackets (a, b, c), then the first number a is from the x plane, the second b is from y, the third c is from z. First take the first coordinate a and mark it on the x axis, left and down if a is positive, right and up if it is negative. Call the resulting letter B.
Then set aside last number c up along the z-axis if it is positive, and down along the same axis if negative. Mark the received point letter D.
From the obtained points, draw projections of the desired point on the planes. That is, at point B, draw two straight lines that will be parallel to the oh and oz axes, at point C, draw straight lines parallel to the ox and oz axes, at point D - straight lines parallel to ox and oz.
If one of the coordinates of a point is zero, the point lies in one of the projection planes. In this case, simply mark the known coordinates on the plane and find point intersection of their projections. Be careful when plotting points with coordinates(a, 0, c) and (a, b, 0), do not forget that the projection onto the x axis is carried out at an angle of 45⁰.
Video on the topic
Sources:
- build by coordinates
Tip 2: How to check that points do not lie on the same line
Based on the axiom describing the properties straight: whatever the straight line is, there is points belonging and not belonging to her. Therefore, it is quite logical that not all points will lie on one straight lines.
You will need
- - pencil;
- - ruler;
- - pen;
- - notebook;
- - calculator.
Instructions
In the event that (x - x1) * (y2 - y1) - (x2 - x1) * (y - y1) will be less than zero, point K is located above or to the left of the line. In other words, only if an equation of the form (x - x1) * (y2 - y1) - (x2 - x1) * (y - y1) = 0 is true, points A, B and K will be located on the same straight.
In other cases, only two points(A and B), which, according to the conditions of the task, lie on straight, will belong to it: the line will not pass through the third point (point K).
Consider a second affiliation option points prime: this time you need to check whether point C(x,y) belongs to the segment with end points B(x1,y1) and A(x2,y2), which is part straight z.
Describe the points of the segment under consideration by the equation pOB+(1-p)OA=z, provided that 0≤p≤1. OB and OA are vectors. If there is a number p that is greater than or equal to 0, but less than or equal to 1, then pOB+(1-p)OA=C, and point C will lie on the segment AB. Otherwise, given point will not belong to this segment.
Write down the equality pOB+(1-p)OA=C coordinate-wise: px1+(1-p)x2=x and py1+(1-p)y2=y.
Find the number p from the first and substitute its value into the second equality. If the equality corresponds to the conditions 0≤p≤1, then point C belongs to the segment AB.
note
Make sure your calculations are correct!
To find k - the slope of a line, you need (y2 - y1)/(x2 - x1).
Sources:
- Algorithm for checking whether a point belongs to a polygon. Ray tracing method in 2019
Three-dimensional space consists of three basic concepts that you gradually learn in school curriculum: point, straight line, plane. When working with some mathematical quantities, you may need to combine these elements, for example, construct a plane in space using a point and a line.
Instructions
To understand the algorithm for constructing planes in space, pay attention to some axioms that describe the properties of a plane or planes. First: through three points that do not lie on the same line, a plane passes through, but only one. Therefore, to construct a plane, you only need three points that satisfy the axiom in position.
Second: through any two points there is a straight line, but only one. Accordingly, a plane can be constructed through a straight line and a point not lying on it. If from the opposite: any line contains at least two points through which it passes, if one more point is known, not on this line, a line can be constructed through these three points, as in point one. Each point of this line will belong to the plane.
Third: a plane passes through two intersecting lines, but only one. Intersecting lines can form only one common point. If in space, they will have an infinite number of common points, and therefore form one straight line. When you know two lines that have an intersection point, you can construct at most one plane passing through these lines.
Fourth: through two parallel lines you can draw a plane, but only one. Accordingly, if you know that the lines are parallel, you can draw a plane through them.
Fifth: an infinite number of planes can be drawn through a straight line. All these planes can be considered as a rotation of one plane around a given line, or as an infinite number of planes having one line of intersection.
So, you can construct a plane if you have found all the elements that determine its position in space: three points that do not lie on a line, a line and a point that does not belong to a line, two intersecting or two parallel lines.
Video on the topic
Did you know that the human body is a mini power plant? Each of us produces a small amount of electricity. This happens both in motion and at rest - then the generation of electricity occurs during internal organs, one of which is the heart.
One of the medical tests that can determine the condition of the heart is an ECG. A cardiologist takes an electrocardiogram to find out where the chest how the atria, valves and ventricles work, their shape and whether there are any functional changes. One of the most important indicators of the ECG is the direction of the electrical axis of the heart.
What is the cardiac axis and how to find it?
The cardiac axis (like the earth's axis) cannot be seen or touched. It is determined only with the help of an electrocardiograph, because it records the electrical activity of the heart. When the cells of the heart muscle tense and relax, obeying impulses coming from nervous system, they form electric field, the center of which is the EOS (electrical axis of the heart).
But if you look at the anatomical atlas, you can draw a vertical line that will divide the heart into two equal parts - this is approximately how the axis of the heart is located. From this we can conclude that the EOS coincides with the so-called anatomical axis. Of course, each person is individual, therefore the electrical axis different people may be located differently (for example, if we start from the statistical value, then in a thin person the EOS is located vertically, and in an obese person it is horizontal).
When does the cardiac axis change position?
Having taken an ECG and finding out how the EOS is located, the cardiologist can tell you how it is in the chest, whether the myocardium (heart) is healthy, how nerve impulses pass to different parts of the heart.
If the electrocardiogram shows that the electrical axis is to the right or to the left, this will indicate to the doctor some pathological process. Deviation to the right may lead to suspicions about the incorrect position of the heart (its displacement may be congenital or occur due to expansion of the aorta, the occurrence of neoplasms and other pathologies). In addition, deviation of EOS is a sign of life-threatening conditions: dextrocardia, His bundle block, myocardial infarction (its anterior wall).
If the EOS is significantly deviated in left side, this may be a sign of cardiomyopathy, hypertrophy of some parts of the heart, apical infarction, or a congenital defect.
A number of heart diseases can be asymptomatic for the time being. Therefore, it is so important to periodically undergo a medical examination, one of the components of which is an ECG. After all, the disease is easier to prevent. And heart disease needs treatment mandatory, because they are a direct threat to life.
Mathematics is a rather complex science. While studying it, you have to not only solve examples and problems, but also work with various shapes and even planes. One of the most used in mathematics is the coordinate system on a plane. Proper work Children have been taught with her for more than one year. Therefore, it is important to know what it is and how to work with it correctly.
Let's figure out what it is this system, what actions can be performed with its help, and also learn its main characteristics and features.
Definition of the concept
A coordinate plane is a plane on which a specific coordinate system is specified. Such a plane is defined by two straight lines intersecting at right angles. At the point of intersection of these lines is the origin of coordinates. Each point on the coordinate plane is specified by a pair of numbers called coordinates.
In a school mathematics course, schoolchildren have to work quite closely with a coordinate system - construct figures and points on it, determine which plane a particular coordinate belongs to, as well as determine the coordinates of a point and write or name them. Therefore, let's talk in more detail about all the features of coordinates. But first, let’s touch on the history of creation, and then we’ll talk about how to work on the coordinate plane.
Historical reference
Ideas about creating a coordinate system existed back in the time of Ptolemy. Even then, astronomers and mathematicians were thinking about how to learn to set the position of a point on a plane. Unfortunately, at that time there was no coordinate system known to us, and scientists had to use other systems.
Initially, they specified points using latitude and longitude. For a long time this was one of the most used methods of putting this or that information on a map. But in 1637, Rene Descartes created his own coordinate system, later named after the “Cartesian” one.
Already at the end of the 17th century. The concept of “coordinate plane” has become widely used in the world of mathematics. Despite the fact that several centuries have passed since the creation of this system, it is still widely used in mathematics and even in life.
Examples of a coordinate plane
Before talking about theory, let's give a few illustrative examples coordinate plane so you can visualize it. The coordinate system is primarily used in chess. On the board, each square has its own coordinates - one coordinate is alphabetic, the second is digital. With its help you can determine the position of a particular piece on the board.
The second most striking example is the game beloved by many “ Sea battle" Remember how, when playing, you name a coordinate, for example, B3, thus indicating exactly where you are aiming. At the same time, when placing ships, you specify points on the coordinate plane.
This coordinate system is widely used not only in mathematics, logic games, but also in military affairs, astronomy, physics and many other sciences.
Coordinate axes
As already mentioned, there are two axes in the coordinate system. Let's talk a little about them, as they are of considerable importance.
The first axis is abscissa - horizontal. It is denoted as ( Ox). The second axis is the ordinate, which runs vertically through the reference point and is denoted as ( Oy). It is these two axes that form the coordinate system, dividing the plane into four quarters. The origin is located at the intersection point of these two axes and takes the value 0 . Only if the plane is formed by two axes intersecting perpendicularly and having a reference point, is it a coordinate plane.
Also note that each of the axes has its own direction. Usually, when constructing a coordinate system, it is customary to indicate the direction of the axis in the form of an arrow. In addition, when constructing a coordinate plane, each of the axes is signed.
Quarters
Now let's say a few words about such a concept as quarters of the coordinate plane. The plane is divided into four quarters by two axes. Each of them has its own number, and the planes are numbered counterclockwise.
Each of the quarters has its own characteristics. So, in the first quarter the abscissa and ordinate are positive, in the second quarter the abscissa is negative, the ordinate is positive, in the third both the abscissa and ordinate are negative, in the fourth the abscissa is positive and the ordinate is negative.
By remembering these features, you can easily determine which quarter a particular point belongs to. In addition, this information may be useful to you if you have to do calculations using the Cartesian system.
Working with the coordinate plane
When we have understood the concept of a plane and talked about its quarters, we can move on to such a problem as working with this system, and also talk about how to put points and coordinates of figures on it. On the coordinate plane, this is not as difficult as it might seem at first glance.
First of all, the system itself is built, all important designations are applied to it. Then already work in progress directly with points or shapes. Moreover, even when constructing figures, points are first drawn on the plane, and then the figures are drawn.
Rules for constructing a plane
If you decide to start marking shapes and points on paper, you will need a coordinate plane. The coordinates of the points are plotted on it. In order to construct a coordinate plane, you only need a ruler and a pen or pencil. First, the horizontal x-axis is drawn, then the vertical axis is drawn. It is important to remember that the axes intersect at right angles.
The next mandatory item is applying markings. On each of the axes in both directions, unit segments are marked and labeled. This is done so that you can then work with the plane with maximum convenience.
Mark a point
Now let's talk about how to plot the coordinates of points on the coordinate plane. This is the basics you need to know to successfully place a variety of shapes on a plane, and even mark equations.
When constructing points, you should remember how their coordinates are correctly written. So, usually when specifying a point, two numbers are written in brackets. The first digit indicates the coordinate of the point along the abscissa axis, the second - along the ordinate axis.
The point should be constructed in this way. First mark on the axis Ox specified point, then mark the point on the axis Oy. Next, draw imaginary lines from these designations and find the place where they intersect - this will be the given point.
All you have to do is mark it and sign it. As you can see, everything is quite simple and does not require any special skills.
Place the figure
Now let's move on to the issue of constructing figures on a coordinate plane. In order to construct any figure on the coordinate plane, you should know how to place points on it. If you know how to do this, then placing a figure on a plane is not so difficult.
First of all, you will need the coordinates of the points of the figure. It is according to them that we will apply the ones you have chosen to our coordinate system. Let us consider the application of a rectangle, a triangle and a circle.
Let's start with a rectangle. It's quite easy to apply. First, four points are marked on the plane, indicating the corners of the rectangle. Then all the points are sequentially connected to each other.
Drawing a triangle is no different. The only thing is that it has three angles, which means that three points are marked on the plane, indicating its vertices.
Regarding the circle, you should know the coordinates of two points. The first point is the center of the circle, the second is the point indicating its radius. These two points are plotted on the plane. Then take a compass and measure the distance between two points. The point of the compass is placed at the point marking the center, and a circle is described.
As you can see, there is nothing complicated here either, the main thing is that you always have a ruler and compass at hand.
Now you know how to plot the coordinates of figures. Doing this on the coordinate plane is not as difficult as it might seem at first glance.
conclusions
So, we have looked at one of the most interesting and basic concepts for mathematics that every schoolchild has to deal with.
We have found out that the coordinate plane is a plane formed by the intersection of two axes. With its help, you can set the coordinates of points and draw shapes on it. The plane is divided into quarters, each of which has its own characteristics.
The main skill that should be developed when working with a coordinate plane is the ability to correctly plot given points on it. To do this, you should know the correct location of the axes, the features of the quarters, as well as the rules by which the coordinates of the points are specified.
We hope that the information we presented was accessible and understandable, and was also useful to you and helped you better understand this topic.
Understanding the Coordinate Plane
Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.
Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.
To construct a coordinate plane, you need to draw $2$ perpendicular straight lines, at the end of which the directions “to the right” and “up” are indicated using arrows. Divisions are applied to the lines, and the point of intersection of the lines is the zero mark for both scales.
Definition 1
The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.
Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.
Coordinate plane
Point coordinates
A point on a coordinate plane is defined by two coordinates.
To determine the coordinates of point $A$ on the coordinate plane, you need to draw straight lines through it that will be parallel coordinate axes(highlighted in the figure with a dotted line). The intersection of the line with the x-axis gives the $x$ coordinate of point $A$, and the intersection with the y-axis gives the y-coordinate of point $A$. When writing the coordinates of a point, the $x$ coordinate is first written, and then the $y$ coordinate.
Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.
To plot a point on the coordinate plane, act in reverse order.
Constructing a point at specified coordinates
Example 1
On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$
Solution.
Construction of point $A$:
- put the number $2$ on the $x$ axis and draw a perpendicular line;
- On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.
Construction of point $B$:
- Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
- On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.
Example 2
Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.
Solution.
Construction of point $C$:
- put the number $3$ on the $x$ axis;
- coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.
Construction of point $D$:
- put the number $2$ on the $y$ axis;
- coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.
Note 1
Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.
Example 3
Determine the coordinates of points A, B, C, D.$
Solution.
Let's determine the coordinates of point $A$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. Thus, we obtain that the point $A (1; 3).$
Let's determine the coordinates of point $B$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. We find that point $B (–2; 4).$
Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.
Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$
Example 4
Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$
Solution.
Construction of point $E$:
- put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
- on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
- at the intersection of perpendicular lines we obtain the point $E (–3; –2).$
Construction of point $F$:
- coordinate $y=0$, which means the point lies on the $x$ axis;
- Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$
Construction of point $G$:
- put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
- on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
- at the intersection of perpendicular lines we obtain the point $G(3; 4).$
Construction of point $H$:
- coordinate $x=0$, which means the point lies on the $y$ axis;
- Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$
Construction of point $O$:
- both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).