Instructions

You are given three points. Let us denote them as (x1, y1), (x2, y2), (x3, y3). It is assumed that these points are the vertices of some triangle. The task is to create equations of its sides - more precisely, equations of those lines on which these sides lie. These equations should look like:
y = k1*x + b1;
y = k2*x + b2;
y = k3*x + b3. Thus, you have to find the angular values ​​k1, k2, k3 and the displacements b1, b2, b3.

Find a line passing through the points (x1, y1), (x2, y2). If x1 = x2, then the desired line is vertical and its equation is x = x1. If y1 = y2, then the line is horizontal and its equation is y = y1. In general, these coordinates will not correspond to each other.

Substituting the coordinates (x1, y1), (x2, y2) into the general equation of the straight line, you get a system of two linear equations: k1*x1 + b1 = y1;
k1*x2 + b1 = y2. Subtract one equation from the other and solve the resulting equation for k1: k1*(x2 - x1) = y2 - y1, therefore k1 = (y2 - y1)/(x2 - x1).

Substituting what you found into any of the original equations, find the expression for b1:((y2 - y1)/(x2 - x1))*x1 + b1 = y1;
b1 = y1 - ((y2 - y1)/(x2 - x1))*x1. Since we already know that x2 ≠ x1, we can simplify the expression by multiplying y1 by (x2 - x1)/(x2 - x1). Then for b1 you will get the following expression: b1 = (x1*y2 - x2*y1)/(x2 - x1).

Check if the third of the given points is on the found line. To do this, substitute (x3, y3) into the resulting equation and see if the equality holds. If it is observed, therefore, all three points lie on the same line, and the triangle degenerates into a segment.

In the same way as described above, derive equations for the lines passing through the points (x2, y2), (x3, y3) and (x1, y1), (x3, y3).

The final form of the equations for the sides of a triangle given by the coordinates of the vertices is: (1) y = ((y2 - y1)*x + (x1*y2 - x2*y1))/(x2 - x1);
(2) y = ((y3 - y2)*x + (x2*y3 - x3*y2))/(x3 - x2);
(3) y = ((y3 - y1)*x + (x1*y3 - x3*y1))/(x3 - x1).

To find equations parties triangle, first of all, we must try to solve the question of how to find the equation of a line on a plane if its direction vector s(m, n) and some point M0(x0, y0) belonging to the line are known.

Instructions

Take an arbitrary (variable, floating) point М(x, y) and construct a vector М0M =(x-x0, y-y0) (write also М0M(x-x0, y-y0)), which will obviously be collinear (parallel ) by k s. Then, we can conclude that the coordinates of these vectors are proportional, so we can create a canonical straight line: (x-x0)/m = (y-y0)/n. It is this ratio that will be used in solving the problem.

All further actions are determined based on the method .1st method. A triangle is given by the coordinates of its three vertices, which in school geometry is given by the lengths of its three parties(see Fig. 1). That is, the condition contains points M1(x1, y1), M2(x2, y2), M3(x3, y3). They correspond to their radius vectors) OM1, 0M2 and OM3 with the same coordinates as the points. For getting equations parties s M1M2 requires its direction vector M1M2=OM2 – OM1=M1M2(x2-x1, y2-y1) and any of the points M1 or M2 (here the point with the lower index is taken).

So for parties y M1M2 canonical equation of the line (x-x1)/(x2-x1)=(y-y1)/(y2-y1). Acting purely inductively, we can write equations the rest parties.For parties s М2М3: (x-x2)/(x3-x2)=(y-y2)/(y3-y2). For parties s М1М3: (x-x1)/(x3-x1)=(y-y1)/(y3-y1).

2nd method. The triangle is defined by two points (the same as before M1(x1, y1) and M2(x2, y2)), as well as the unit vectors of the directions of the other two parties. For parties s M2M3: p^0(m1, n1). For M1M3: q^0(m2, n2). Therefore for parties s M1M2 will be the same as in the first method: (x-x1)/(x2-x1)=(y-y1)/(y2-y1).

For parties s М2М3 as a point (x0, y0) of the canonical equations(x1, y1), and the direction vector is p^0(m1, n1). For parties s M1M3, (x2, y2) is taken as the point (x0, y0), the direction vector is q^0(m2, n2). Thus, for M2M3: equation (x-x1)/m1=(y-y1)/n1. For M1M3: (x-x2)/m2=(y-y2)/n2.

Video on the topic

Tip 3: How to find the height of a triangle if the coordinates of the points are given

The height is the straight line segment connecting the top of the figure with the opposite side. This segment must be perpendicular to the side, so only one can be drawn from each vertex height. Since there are three vertices in this figure, there are the same number of heights. If a triangle is given by the coordinates of its vertices, the length of each of the heights can be calculated, for example, by using the formula for finding the area and calculating the lengths of the sides.

Instructions

Start by calculating the side lengths triangle. Designate coordinates figures like this: A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃). Then you can calculate the length of side AB using the formula AB = √((X₁-X₂)² + (Y₁-Y₂)² + (Z₁-Z₂)²). For the other two sides these will look like this: BC = √((X₂-X₃)² + (Y₂-Y₃)² + (Z₂-Z₃)²) and AC = √((X₁-X₃)² + (Y₁-Y₃ )² + (Z₁-Z₃)²). For example, for triangle with coordinates A(3,5,7), B(16,14,19) and C(1,2,13) ​​the length of side AB will be √((3-16)² + (5-14)² + (7 -19)²) = √(-13² + (-9²) + (-12²)) = √(169 + 81 + 144) = √394 ≈ 19.85. The lengths of the sides BC and AC, calculated in the same way, will be √(15² + 12² + 6²) = √405 ≈ 20.12 and √(2² + 3² + (-6²)) = √49 = 7.

Knowing the lengths of the three sides obtained in the previous step is enough to calculate the area triangle(S) according to Heron's formula: S = ¼ * √((AB+BC+CA) * (BC+CA-AB) * (AB+CA-BC) * (AB+BC-CA)). For example, substituting into this formula the values ​​​​obtained from the coordinates triangle-sample from the previous step, this will give the value: S = ¼*√((19.85+20.12+7) * (20.12+7-19.85) * (19.85+7-20.12 ) * (19.85+20.12-7)) = ¼*√(46.97 * 7.27 * 6.73 * 32.97) ≈ ¼*√75768.55 ≈ ¼*275.26 = 68.815 .

Based on area triangle, calculated in the previous step, and the lengths of the sides obtained in the second step, calculate the heights for each of the sides. Since the area is equal to half the product of the height and the length of the side to which it is drawn, to find the height, divide the doubled area by the length of the desired side: H = 2*S/a. For the example used above, the height lowered to side AB will be 2*68.815/16.09 ≈ 8.55, the height to side BC will have a length of 2*68.815/20.12 ≈ 6.84, and for side AC this value will be equal to 2 *68.815/7 ≈ 19.66.

Sources:

• given points find the area of ​​the triangle

Tip 4: How to use the coordinates of the vertices of a triangle to find the equations of its sides

In analytical geometry, a triangle on a plane can be defined in a Cartesian coordinate system. Knowing the coordinates of the vertices, you can create equations for the sides of the triangle. These will be the equations of three straight lines, which, intersecting, form a figure.

How to learn to solve problems in analytical geometry?
Typical problem with a triangle on a plane

This lesson is created on the approach to the equator between the geometry of the plane and the geometry of space. At the moment, there is a need to systematize the accumulated information and answer a very important question: how to learn to solve problems in analytical geometry? The difficulty is that you can come up with an infinite number of problems in geometry, and no textbook will contain all the multitude and variety of examples. Is not derivative of a function with five rules of differentiation, a table and several techniques….

There is a solution! I will not speak loudly about the fact that I have developed some kind of grandiose technique, however, in my opinion, there is an effective approach to the problem under consideration, which allows even a complete dummy to achieve good and excellent results. At least, the general algorithm for solving geometric problems took shape very clearly in my head.

WHAT YOU NEED TO KNOW AND BE ABLE TO DO
for successfully solving geometry problems?

There is no escape from this - in order not to randomly poke buttons with your nose, you need to master the basics of analytical geometry. Therefore, if you have just started studying geometry or have completely forgotten it, please start with the lesson Vectors for dummies. In addition to vectors and actions with them, you need to know the basic concepts of plane geometry, in particular, equation of a line in a plane And . The geometry of space is presented in articles Plane equation, Equations of a line in space, Basic problems on a straight line and a plane and some other lessons. Curved lines and spatial surfaces of the second order stand somewhat apart, and there are not so many specific problems with them.

Let's assume that the student already has basic knowledge and skills in solving the simplest problems of analytical geometry. But it happens like this: you read the statement of the problem, and... you want to close the whole thing altogether, throw it in the far corner and forget it, like a bad dream. Moreover, this fundamentally does not depend on the level of your qualifications; from time to time I myself come across tasks for which the solution is not obvious. What to do in such cases? There is no need to be afraid of a task that you don’t understand!

Firstly, should be installed - Is this a “flat” or spatial problem? For example, if the condition includes vectors with two coordinates, then, of course, this is the geometry of a plane. And if the teacher loaded the grateful listener with a pyramid, then there is clearly the geometry of space. The results of the first step are already quite good, because we managed to cut off a huge amount of information unnecessary for this task!

Second. The condition will usually concern you with some geometric figure. Indeed, walk along the corridors of your native university, and you will see a lot of worried faces.

In “flat” problems, not to mention the obvious points and lines, the most popular figure is a triangle. We will analyze it in great detail. Next comes the parallelogram, and much less common are the rectangle, square, rhombus, circle, and other shapes.

In spatial problems, the same flat figures + the planes themselves and common triangular pyramids with parallelepipeds can fly.

Question two - Do you know everything about this figure? Suppose the condition talks about an isosceles triangle, and you very vaguely remember what kind of triangle it is. We open a school textbook and read about an isosceles triangle. What to do... the doctor said a rhombus, that means a rhombus. Analytical geometry is analytical geometry, but the problem will be solved by the geometric properties of the figures themselves, known to us from the school curriculum. If you don’t know what the sum of the angles of a triangle is, you can suffer for a long time.

Third. ALWAYS try to follow the drawing(on a draft/finish copy/mentally), even if this is not required by the condition. In “flat” problems, Euclid himself ordered to pick up a ruler and a pencil - and not only in order to understand the condition, but also for the purpose of self-test. In this case, the most convenient scale is 1 unit = 1 cm (2 notebook cells). Let's not talk about careless students and mathematicians spinning in their graves - it is almost impossible to make a mistake in such problems. For spatial tasks, we perform a schematic drawing, which will also help analyze the condition.

A drawing or schematic drawing often allows you to immediately see the way to solve a problem. Of course, for this you need to know the foundation of geometry and understand the properties of geometric shapes (see the previous paragraph).

Fourth. Development of a solution algorithm. Many geometry problems are multi-step, so the solution and its design is very convenient to break down into points. Often the algorithm immediately comes to mind after you read the condition or complete the drawing. In case of difficulties, we start with the QUESTION of the task. For example, according to the condition “you need to construct a straight line...”. Here the most logical question is: “What is enough to know to construct this straight line?” Suppose, “we know the point, we need to know the direction vector.” We ask the following question: “How to find this direction vector? Where?" etc.

Sometimes there is a “bug” - the problem is not solved and that’s it. The reasons for the stop may be the following:

– Serious gap in basic knowledge. In other words, you do not know and/or do not see some very simple thing.

– Ignorance of the properties of geometric figures.

- The task was difficult. Yes, it happens. There is no point in steaming for hours and collecting tears in a handkerchief. Seek advice from your teacher, fellow students, or ask a question on the forum. Moreover, it is better to make its statement concrete - about that part of the solution that you do not understand. A cry in the form of “How to solve the problem?” doesn't look very good... and, above all, for your own reputation.

Stage five. We decide-check, decide-check, decide-check-give an answer. It is beneficial to check each point of the task immediately after it is completed. This will help you spot the error immediately. Naturally, no one forbids quickly solving the entire problem, but there is a risk of rewriting everything again (often several pages).

These are, perhaps, all the main considerations that should be followed when solving problems.

The practical part of the lesson is presented in plane geometry. There will be only two examples, but it won’t seem enough =)

Let's go through the thread of the algorithm that I just looked at in my little scientific work:

Example 1

Three vertices of a parallelogram are given. Find the top.

Let's start to understand:

Step one: It is obvious that we are talking about a “flat” problem.

Step two: The problem deals with a parallelogram. Does everyone remember this parallelogram figure? There is no need to smile, many people receive their education at 30-40-50 or more years old, so even simple facts can be erased from memory. The definition of a parallelogram is found in Example No. 3 of the lesson Linear (non) dependence of vectors. Basis of vectors.

Step three: Let's make a drawing on which we mark three known vertices. It’s funny that it’s not difficult to immediately construct the desired point:

Constructing it is, of course, good, but the solution must be formulated analytically.

Step four: Development of a solution algorithm. The first thing that comes to mind is that a point can be found as the intersection of lines. We do not know their equations, so we will have to deal with this issue:

1) Opposite sides are parallel. By points Let's find the direction vector of these sides. This is the simplest problem that was discussed in class. Vectors for dummies.

Note: it is more correct to say “the equation of a line containing a side,” but here and further for brevity I will use the phrases “equation of a side,” “direction vector of a side,” etc.

3) Opposite sides are parallel. Using the points, we find the direction vector of these sides.

4) Let’s create an equation of a straight line using a point and a direction vector

In paragraphs 1-2 and 3-4, we actually solved the same problem twice; by the way, it was discussed in example No. 3 of the lesson The simplest problems with a straight line on a plane. It was possible to take a longer route - first find the equations of the lines and only then “pull out” the direction vectors from them.

5) Now the equations of the lines are known. All that remains is to compose and solve the corresponding system of linear equations (see examples No. 4, 5 of the same lesson The simplest problems with a straight line on a plane).

The point has been found.

The problem is quite simple and its solution is obvious, but there is a shorter way!

Second solution:

The diagonals of a parallelogram are bisected by their point of intersection. I marked the point, but so as not to clutter the drawing, I did not draw the diagonals themselves.

Let's compose the equation of the side point by point :

To check, you should mentally or on a draft substitute the coordinates of each point into the resulting equation. Now let's find the slope. To do this, we rewrite the general equation in the form of an equation with a slope coefficient:

Thus, the slope is:

Similarly, we find the equations of the sides. I don’t see much point in describing the same thing, so I’ll immediately give the finished result:

2) Find the length of the side. This is the simplest problem covered in class. Vectors for dummies. For points we use the formula:

Using the same formula it is easy to find the lengths of other sides. The check can be done very quickly with a regular ruler.

We use the formula .

Let's find the vectors:

Thus:

By the way, along the way we found the lengths of the sides.

As a result:

Well, it looks like the truth; to make it more convincing, you can attach a protractor to the corner.

Attention! Do not confuse the angle of a triangle with the angle between straight lines. The angle of a triangle can be obtuse, but the angle between straight lines cannot (see the last paragraph of the article The simplest problems with a straight line on a plane). However, to find the angle of a triangle, you can also use the formulas from the above lesson, but the roughness is that those formulas always give an acute angle. With their help, I solved this problem in draft and got the result. And on the final copy I would have to write down additional excuses, that .

4) Write an equation for a line passing through a point parallel to the line.

Standard task, discussed in detail in example No. 2 of the lesson The simplest problems with a straight line on a plane. From the general equation of the line Let's take out the guide vector. Let's create an equation of a straight line using a point and a direction vector:

How to find the height of a triangle?

5) Let’s create an equation for the height and find its length.

There is no escape from strict definitions, so you’ll have to steal from a school textbook:

Triangle height is called the perpendicular drawn from the vertex of the triangle to the line containing the opposite side.

That is, it is necessary to create an equation for a perpendicular drawn from the vertex to the side. This task is discussed in examples No. 6, 7 of the lesson The simplest problems with a straight line on a plane. From Eq. remove the normal vector. Let's compose the height equation using a point and a direction vector:

Please note that we do not know the coordinates of the point.

Sometimes the height equation is found from the ratio of the angular coefficients of perpendicular lines: . In this case, then: . Let's compose the height equation using a point and an angular coefficient (see the beginning of the lesson Equation of a straight line on a plane):

The height length can be found in two ways.

There is a roundabout way:

a) find – the point of intersection of height and side;
b) find the length of the segment using two known points.

But in class The simplest problems with a straight line on a plane a convenient formula for the distance from a point to a line was considered. The point is known: , the equation of the line is also known: , Thus:

6) Calculate the area of ​​the triangle. In space, the area of ​​a triangle is traditionally calculated using vector product of vectors, but here we are given a triangle on a plane. We use the school formula:
– The area of ​​a triangle is equal to half the product of its base and its height.

In this case:

How to find the median of a triangle?

7) Let's create an equation for the median.

Median of a triangle called a segment connecting the vertex of a triangle with the middle of the opposite side.

a) Find the point - the middle of the side. We use formulas for the coordinates of the midpoint of a segment. The coordinates of the ends of the segment are known: , then the coordinates of the middle:

Thus:

Let's compose the median equation point by point :

To check the equation, you need to substitute the coordinates of the points into it.

8) Find the point of intersection of the height and the median. I think everyone has already learned how to perform this element of figure skating without falling:

An example of solving some tasks from the standard work “Analytical geometry on a plane”

The vertices are given,
,
triangle ABC. Find:

Equations of all sides of a triangle;

System of linear inequalities defining a triangle ABC;

Equations of altitude, median and bisector of a triangle drawn from the vertex A;

The intersection point of the triangle's altitudes;

The intersection point of the triangle's medians;

Length of the height lowered to the side AB;

Corner A;

Make a drawing.

Let the vertices of the triangle have coordinates: A (1; 4), IN (5; 3), WITH(3; 6). Let's draw a drawing right away:

1. To write down the equations of all sides of a triangle, we use the equation of a straight line passing through two given points with coordinates ( x 0 , y 0 ) And ( x 1 , y 1 ):

=

Thus, substituting instead of ( x 0 , y 0 ) point coordinates A, and instead of ( x 1 , y 1 ) point coordinates IN, we get the equation of the line AB:

The resulting equation will be the equation of the straight line AB, written in general form. Similarly, we find the equation of the straight line AC:

And also the equation of the straight line Sun:

2. Note that the set of points of the triangle ABC represents the intersection of three half-planes, and each half-plane can be defined using a linear inequality. If we take the equation of either side ∆ ABC, For example AB, then the inequalities

And

define points lying on opposite sides of a line AB. We need to choose the half-plane where point C lies. Let’s substitute its coordinates into both inequalities:

The second inequality will be correct, which means that the required points are determined by the inequality

.

We do the same with straight line BC, its equation
. We use point A (1, 1) as a test point:

This means that the required inequality has the form:

.

If we check straight line AC (test point B), we get:

This means that the required inequality will have the form

We finally obtain a system of inequalities:

The signs “≤”, “≥” mean that points lying on the sides of the triangle are also included in the set of points that make up the triangle ABC.

3. a) In order to find the equation of the height dropped from the vertex A to the side Sun, consider the equation of the side Sun:
.
Vector with coordinates Sun perpendicular to the side A and therefore parallel to the height. Let's write down the equation of a straight line passing through a point
:

parallel to the vector A This is the equation for the height omitted from t. Sun.

to the side Sun b) Find the coordinates of the middle of the side

according to the formulas:
Here IN- these are the coordinates of t.
, A WITH– coordinates t.

. Let's substitute and get: A The straight line passing through this point and the point

is the required median:
c) We will look for the equation of the bisector based on the fact that in an isosceles triangle the height, median and bisector descended from one vertex to the base of the triangle are equal. Let's find two vectors
And

and their lengths:
Then the vector
has the same direction as the vector
, and its length
Likewise, the unit vector
coincides in direction with the vector

Sum of vectors A is a vector that coincides in direction with the bisector of the angle

. Thus, the equation of the desired bisector can be written as: IN 4) We have already constructed the equation for one of the heights. Let's construct an equation for another height, for example, from the vertex AC. Side
given by the equation
So the vector AC perpendicular IN, and thus parallel to the desired height. Then the equation of the line passing through the vertex
in the direction of the vector AC(i.e. perpendicular

), has the form:

- coordinates of this point.

5. Middle AB has coordinates
. Let us write the equation of the median to the side AB. This line passes through points with coordinates (3, 2) and (3, 6), which means its equation has the form:

Note that a zero in the denominator of a fraction in the equation of a straight line means that this straight line runs parallel to the ordinate axis.

To find the intersection point of the medians, it is enough to solve the system of equations:

The intersection point of the medians of a triangle has coordinates
.

6. Length of height lowered to the side AB, equal to the distance from the point WITH to a straight line AB with equation
and is found by the formula:

7. Cosine of angle A can be found using the formula for the cosine of the angle between vectors c) We will look for the equation of the bisector based on the fact that in an isosceles triangle the height, median and bisector descended from one vertex to the base of the triangle are equal. Let's find two vectors , which is equal to the ratio of the scalar product of these vectors to the product of their lengths:

.