Problem 1. The coordinates of the vertices of triangle ABC are given: A(4; 3), B(16;-6), C(20; 16). Find: 1) the length of side AB; 2) equations of sides AB and BC and their angular coefficients; 3) angle B in radians with an accuracy of two digits; 4) equation of height CD and its length; 5) the equation of the median AE and the coordinates of the point K of the intersection of this median with the height CD; 6) the equation of a straight line passing through point K parallel to side AB; 7) coordinates of point M, located symmetrically to point A relative to straight line CD.

Solution:

1. The distance d between points A(x 1 ,y 1) and B(x 2 ,y 2) is determined by the formula

Applying (1), we find the length of side AB:

2. The equation of the line passing through the points A(x 1 ,y 1) and B(x 2 ,y 2) has the form

(2)

Substituting the coordinates of points A and B into (2), we obtain the equation of side AB:

Having solved the last equation for y, we find the equation of side AB in the form of a straight line equation with an angular coefficient:

where

Substituting the coordinates of points B and C into (2), we obtain the equation of straight line BC:

Or

3. It is known that the tangent of the angle between two straight lines, the angular coefficients of which are respectively equal, is calculated by the formula

(3)

The desired angle B is formed by straight lines AB and BC, the angular coefficients of which are found: Applying (3), we obtain

Or glad.

4. The equation of a straight line passing through a given point in a given direction has the form

(4)

The height CD is perpendicular to side AB. To find the slope of the height CD, we use the condition of perpendicularity of the lines. Since then Substituting into (4) the coordinates of point C and the found angular coefficient of height, we obtain

To find the length of the height CD, we first determine the coordinates of point D - the point of intersection of straight lines AB and CD. Solving the system together:

we find those. D(8;0).

Using formula (1) we find the length of the height CD:

5. To find the equation of the median AE, we first determine the coordinates of point E, which is the middle of side BC, using the formulas for dividing a segment into two equal parts:

(5)

Hence,

Substituting the coordinates of points A and E into (2), we find the equation for the median:

To find the coordinates of the point of intersection of the height CD and the median AE, we solve together the system of equations

We find.

6. Since the desired straight line is parallel to side AB, its angular coefficient will be equal to the angular coefficient of straight line AB. Substituting into (4) the coordinates of the found point K and the angular coefficient we obtain

3x + 4y – 49 = 0 (KF)

7. Since the straight line AB is perpendicular to the straight line CD, the desired point M, located symmetrically to the point A relative to the straight line CD, lies on the straight line AB. In addition, point D is the midpoint of segment AM. Using formulas (5), we find the coordinates of the desired point M:

Triangle ABC, height CD, median AE, straight line KF and point M are constructed in the xOy coordinate system in Fig. 1.

Task 2. Create an equation for the locus of points whose distances to a given point A(4; 0) and to a given line x=1 are equal to 2.

Solution:

In the xOy coordinate system, we construct the point A(4;0) and the straight line x = 1. Let M(x;y) be an arbitrary point of the desired geometric location of points. Let us lower the perpendicular MB to the given line x = 1 and determine the coordinates of point B. Since point B lies on the given line, its abscissa is equal to 1. The ordinate of point B is equal to the ordinate of point M. Therefore, B(1;y) (Fig. 2 ).

According to the conditions of the problem |MA|: |MV| = 2. Distances |MA| and |MB| we find from formula (1) of problem 1:

Squaring the left and right sides, we get

or

The resulting equation is a hyperbola in which the real semi-axis is a = 2, and the imaginary half-axis is

Let's define the foci of a hyperbola. For a hyperbola, the equality is satisfied. Therefore, and – hyperbole tricks. As you can see, the given point A(4;0) is the right focus of the hyperbola.

Let us determine the eccentricity of the resulting hyperbola:

The equations of the hyperbola asymptotes have the form and . Therefore, or and are asymptotes of a hyperbola. Before constructing a hyperbola, we construct its asymptotes.

Problem 3. Create an equation for the locus of points equidistant from the point A(4; 3) and the straight line y = 1. Reduce the resulting equation to its simplest form.

Solution: Let M(x; y) be one of the points of the desired geometric locus of points. Let us drop the perpendicular MB from point M to this straight line y = 1 (Fig. 3). Let us determine the coordinates of point B. Obviously, the abscissa of point B is equal to the abscissa of point M, and the ordinate of point B is equal to 1, i.e. B(x; 1). According to the conditions of the problem |MA|=|MV|. Consequently, for any point M(x;y) belonging to the desired geometric locus of points, the following equality is true:

The resulting equation defines a parabola with a vertex at the point. To bring the parabola equation to its simplest form, let us set and y + 2 = Y, then the parabola equation takes the form:

Exercise. Points A (2,1), B (1,-2), C (-1,0) are the vertices of triangle ABC.
a) Find the equations of the sides of triangle ABC.
b) Find the equation of one of the medians of triangle ABC.
c) Find the equation of one of the altitudes of triangle ABC.
d) Find the equation of one of the bisectors of triangle ABC.
e) Find the area of ​​triangle ABC.

Solution We do it using a calculator.
The coordinates of the triangle are given: A(2,1), B(1,-2), C(-1,0).
1) Vector coordinates
We find the coordinates of the vectors using the formula:
X = x j - x i ; Y = y j - y i

For example, for vector AB

X = 1-2 = -1; Y = -2-1 = -3
AB(-1;-3)
AC(-3;-1)
BC(-2;2)
2) Vector modules

3) Angle between straight lines
The angle between vectors a 1 (X 1 ;Y 1), a 2 (X 2 ;Y 2) can be found using the formula:

where a 1 a 2 = X 1 X 2 + Y 1 Y 2
Find the angle between sides AB and AC

γ = arccos(0.6) = 53.13 0
4) Vector projection
Vector projection b to vector a can be found using the formula:

Let's find the projection of vector AB onto vector AC

5) Area of ​​the triangle



Solution


Using the formula we get:

6) Division of a segment in this relation
The radius vector r of point A, dividing the segment AB in the ratio AA:AB = m 1:m 2, is determined by the formula:

The coordinates of point A are found using the formulas:




Equation of the median of a triangle
Let us denote the middle of side BC by the letter M. Then we will find the coordinates of point M using the formulas for dividing a segment in half.


M(0;-1)
We find the equation of the median AM using the formula for the equation of a straight line passing through two given points. The median AM passes through points A(2;1) and M(0;-1), therefore:

or

or
y = x -1 or y -x +1 = 0
7) Equation of a line


Equation of line AB

or

or
y = 3x -5 or y -3x +5 = 0
Equation of line AC

or

or
y = 1 / 3 x + 1 / 3 or 3y -x - 1 = 0
Equation of line BC

or

or
y = -x -1 or y + x +1 = 0
8) Length of the altitude of the triangle drawn from vertex A
The distance d from the point M 1 (x 1 ;y 1) to the straight line Ax + By + C = 0 is equal to the absolute value of the quantity:

Find the distance between point A(2;1) and line BC (y + x +1 = 0)

9) Equation of height through vertex C
The straight line passing through the point M 0 (x 0 ;y 0) and perpendicular to the straight line Ax + By + C = 0 has a direction vector (A;B) and, therefore, is represented by the equations:


This equation can be found in another way. To do this, let's find the slope k 1 of straight line AB.
AB equation: y = 3x -5, i.e. k 1 = 3
Let's find the angular coefficient k of the perpendicular from the condition of perpendicularity of two straight lines: k 1 *k = -1.
Substituting the slope of this line instead of k 1, we get:
3k = -1, whence k = -1 / 3
Since the perpendicular passes through the point C(-1,0) and has k = -1 / 3, we will look for its equation in the form: y-y 0 = k(x-x 0).
Substituting x 0 = -1, k = -1 / 3, y 0 = 0 we get:
y-0 = -1 / 3 (x-(-1))
or
y = -1/3 x - 1/3
Triangle bisector equation
Let us find the bisector of angle A. Let us denote the point of intersection of the bisector with side BC as M.
Let's use the formula:

AB equation: y -3x +5 = 0, AC equation: 3y -x - 1 = 0

^A ≈ 53 0
The bisector divides the angle in half, therefore the angle NAK ≈ 26.5 0
The slope of AB is equal to 3 (since y -3x +5 = 0). The angle of inclination is 72
^NKA≈ 180 0 - 72 0 = 108 0
^ANK ≈ 180 0 - (108 0 + 26.5 0) ≈ 45.5 0
tg(45.5 0) = 1
The bisector passes through the point A(2,1), using the formula, we have:
y - y 0 = k(x - x 0)
y - 1 = 1(x - 2)
or
y=x-1
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Example. The coordinates of the vertices of triangle ABC are given: A(–3; –1), B(4; 6), C(8; –2).
Required: 1) calculate the length of the side of the aircraft; 2) create an equation for the side BC; 3) find the interior angle of the triangle at vertex B; 4) compose an equation for the height AK drawn from vertex A; 5) find the coordinates of the center of gravity of a homogeneous triangle (the intersection points of its medians); 6) make a drawing in a coordinate system.

Exercise. The coordinates of the vertices of triangle ABC are given: A(7;4), B(-9;-8), C(-2;16). Required:

1. write an equation for the median drawn from vertex B and calculate its length.
2. write an equation for the height drawn from vertex A and calculate its length.
3. find the cosine of interior angle B of triangle ABC.

Make a drawing.

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Example No. 3. Given vertices A(1;1), B(7;4), C(4;5) of a triangle. Find: 1) the length of side AB; 2) internal angle A in radians with an accuracy of 0.001. Make a drawing.
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Example No. 4. Given vertices A(1;1), B(7;4), C(4;5) of a triangle. Find: 1) the equation of the height drawn through the vertex C; 2) the equation of the median drawn through the vertex C; 3) the point of intersection of the triangle’s altitudes; 4) the length of the height lowered from vertex C. Make a drawing.
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Example No. 5. Given the vertices of triangle ABC: A(-5;0), B(7;-9), C(11;13). Determine: 1) the length of side AB; 2) equation of sides AB and AC and their angular coefficients; 3) area of ​​the triangle.

We find the coordinates of the vectors using the formula: X = x j - x i ; Y = y j - y i
here X,Y coordinates of the vector; x i, y i - coordinates of point A i; x j, y j - coordinates of point A j
For example, for vector AB
X = x 2 - x 1 ; Y = y 2 - y 1
X = 7-(-5) = 12; Y = -9-0 = -9
AB(12;-9), AC(16;13), BC(4;22).

Length of sides of triangle
The length of the vector a(X;Y) is expressed through its coordinates by the formula:


Area of ​​a triangle
Let the points A 1 (x 1 ; y 1), A 2 (x 2 ; y 2), A 3 (x 3 ; y 3) be the vertices of the triangle, then its area is expressed by the formula:

On the right side there is a second-order determinant. The area of ​​a triangle is always positive.
Solution. Taking A as the first vertex, we find:

Using the formula we get:

Equation of a line
A straight line passing through points A 1 (x 1 ; y 1) and A 2 (x 2 ; y 2) is represented by the equations:

Equation of line AB
Canonical equation of the line:

or

or
y = -3 / 4 x -15 / 4 or 4y + 3x +15 = 0
The slope of straight line AB is equal to k = -3 / 4
Equation of line AC

or

or
y = 13 / 16 x + 65 / 16 or 16y -13x - 65 = 0
The slope of straight line AB is equal to k = 13 / 16

Exercise. The coordinates of the vertices of the ABCD pyramid are given. Required:

1. Write the vectors in the ort system and find the modules of these vectors.
2. Find the angle between the vectors.
3. Find the projection of a vector onto a vector.
4. Find the area of ​​face ABC.
5. Find the volume of the pyramid ABCD.

Solution
Example No. 1
A 1 (1,8,2), A 2 (5,2,6), A 3 (0,-1,-2), A 4 (-2,3,-1): Example No. 2
A 1 (5,2,1), A 2 (-3,9,3), A 3 (-1,3,5), A 4 (-1,-5,2): Example No. 3
A 1 (-1,0,2), A 2 (-2,0,6), A 3 (-3,1,2), A 4 (-1,2,4): Example No. 4

Exercise. Find the acute angle between the lines x + y -5 = 0 and x + 4y - 8 = 0.
Recommendations for solution. The problem is solved using the service Angle between two straight lines.
Answer: 30.96 o

Example No. 1. The coordinates of points A1(1;0;2), A2(2;1;1), A3(-1;2;0), A4(-2;-1;-1) are given. Find the length of edge A1A2. Create an equation for edge A1A4 and face A1A2A3. Compose an equation for the height lowered from point A4 to the plane A1A2A3. Find the area of ​​triangle A1A2A3. Find the volume of the triangular pyramid A1A2A3A4.

We find the coordinates of the vectors using the formula: X = x j - x i ; Y = y j - y i ; Z = z j - z i
here X,Y,Z coordinates of the vector; x i, y i, z i - coordinates of point A i; x j, y j, z j - coordinates of point A j;
So, for the vector A 1 A 2 they will be as follows:
X = x 2 - x 1 ; Y = y 2 - y 1 ; Z = z 2 - z 1
X = 2-1; Y = 1-0; Z = 1-2
A 1 A 2 (1;1;-1)
A 1 A 3 (-2;2;-2)
A 1 A 4 (-3;-1;-3)
A 2 A 3 (-3;1;-1)
A 2 A 4 (-4;-2;-2)
A 3 A 4 (-1;-3;-1)
The length of the vector a(X;Y;Z) is expressed through its coordinates by the formula:

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 the 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 of age, 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 task 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 in order 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 seems to be true; to be 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 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: