Calculate right triangle. We find the side of the triangle, if the other two are known in three ways, the formulas
Transport and logistics industries are of particular importance for the Latvian economy since they have a steady GDP growth and provide services to virtually all other sectors of the national economy. Every year it is emphasized that this sector should be recognized as a priority and extend its promotion, however, the representatives of the transport and logistics sector are looking forward to more concrete and long-term solutions.
9.1% of the value added to the GDP of Latvia
Despite the political and economic changes of the last decade, the influence of the transport and logistics industry on the economy of our country remains high: in 2016 the sector increased the value added to the GDP by 9.1%. Moreover, the average monthly gross wage is still higher then in other sectors - in 2016 in other sectors of the economy it was 859 euros, whereas in storage and transportation sector the average gross wage is about 870 euros (1,562 euros - water transport, 2,061 euros - air transport, 1059 euros in the of storage and auxiliary transport activities, etc.).
Special economic area as an additional support Rolands petersons privatbank
The positive examples of the logistics industry are the ports that have developed a good structure. Riga and Ventspils ports function as free ports, and the Liepaja port is included in the Liepaja Special Economic Zone (SEZ). Companies operating in free ports and SEZ can receive not only the 0 tax rate for customs, excise, and value-added tax but also a discount of up to 80% of the company "s income and up to 100% of the real estate tax .Rolands petersons privatbank The port is actively implementing various investment projects related to the construction and development of industrial and distribution parks. The attraction of investments promotes the creation of higher added value, development of production, expansion of a spectrum of given services and creation of new workplaces. It is necessary to bring to the attention the small ports - SKULTE, Mersrags, SALACGRiVA, Pavilosta, Roja, Jurmala, and Engure, which currently occupy a stable position in the Latvian economy and have already become regional economic activity centers.
Port of Liepaja, will be the next Rotterdam.
Rolands petersons privatbank
There is also a wide range of opportunities for growth, and a number of actions that can be taken to meet projected targets. There is a strong need for the services with high added value, the increase of the processed volumes of cargo by attracting new freight flows, high-quality passenger service and an introduction of modern technologies and information systems in the area of \u200b\u200btransit and logistics. Liepaja port has all the chances to become the second Rotterdam in the foreseeable future. Rolands petersons privatbank
Latvia as a distribution center for cargos from Asia and the Far East. Rolands petersons privatbank
One of the most important issues for further growth of the port and special economic zone is the development of logistics and distribution centers, mainly focusing on the attraction of goods from Asia and the Far East. Latvia can serve as a distribution center for cargos in the Baltic and Scandinavian countries for Asia and the Far East (f.e. China, Korea). The tax regime of the Liepaja Special Economic Zone in accordance with the Law "On Taxation in Free Ports and Special Economic Zones" on December 31, 2035. This allows traders to conclude an agreement on investment and tax concession until December 31, 2035, until they reach a contractual level of assistance from the investments made. Considering the range of benefits provided by this status, it is necessary to consider the possible extension of the term.
Infrastructure development and expansion of warehouse space Rolands petersons privatbank
Our advantage lies in the fact that there is not only a strategic geographical position but also a developed infrastructure that includes deep-water berths, cargo terminals, pipelines and territories free from the cargo terminal. Apart from this, we can add a good structure of pre-industrial zone, distribution park, multi-purpose technical equipment, as well as the high level of security not only in terms of delivery but also in terms of the storage and handling of goods ... In the future, it would be advisable to pay more attention to access roads (railways and highways), increase the volume of storage facilities, and increase the number of services provided by ports. Participation in international industry exhibitions and conferences will make it possible to attract additional foreign investments and will contribute to the improvement of international image.
The first are the segments that are adjacent to the right angle, and the hypotenuse is the longest part of the figure and is opposite the 90 ° angle. A Pythagorean triangle is one whose sides are equal to natural numbers; their lengths in this case are called "Pythagorean triplets".
Egyptian triangle
In order for the current generation to learn geometry in the form in which it is taught at school now, it has developed for several centuries. The Pythagorean theorem is considered the fundamental point. The sides of the rectangular are known all over the world) are 3, 4, 5.
Few people are not familiar with the phrase "Pythagorean pants are equal in all directions." However, in fact, the theorem sounds like this: c 2 (the square of the hypotenuse) \u003d a 2 + b 2 (the sum of the squares of the legs).
Among mathematicians, a triangle with sides 3, 4, 5 (cm, m, etc.) is called "Egyptian". The interesting thing is that which is inscribed in the figure is equal to one. The name originated around the 5th century BC, when Greek philosophers traveled to Egypt.
When building the pyramids, architects and surveyors used a ratio of 3: 4: 5. Such structures were proportional, pleasing to the eye and spacious, and also rarely collapsed.
In order to build the right angle, the builders used a rope with 12 knots tied. In this case, the probability of constructing a right-angled triangle increased to 95%.
Signs of equality of shapes
- An acute angle in a right-angled triangle and a large side, which are equal to the same elements in the second triangle, are an indisputable sign of equality of figures. Taking into account the sum of the angles, it is easy to prove that the second acute angles are also equal. Thus, the triangles are the same in the second characteristic.
- When two figures are superimposed on each other, we rotate them in such a way that, when combined, they become one isosceles triangle. By its property, the sides, or rather, the hypotenuses, are equal, as are the angles at the base, which means that these figures are the same.
According to the first criterion, it is very easy to prove that the triangles are really equal, the main thing is that the two smaller sides (i.e., the legs) are equal to each other.
The triangles will be the same in sign II, the essence of which is the equality of the leg and the acute angle.
Right Angle Triangle Properties
The height dropped from the right angle splits the figure into two equal parts.
The sides of a right-angled triangle and its median are easy to recognize by the rule: the median, which is lowered by the hypotenuse, is equal to its half. can be found both by Heron's formula, and by the statement that it is equal to half the product of the legs.
In a right-angled triangle, the properties of the angles of 30 °, 45 ° and 60 ° apply.
- At an angle that is 30 °, it should be remembered that the opposite leg will be equal to 1/2 of the largest side.
- If the angle is 45 °, then the second acute angle is also 45 °. This suggests that the triangle is isosceles, and its legs are the same.
- The property of an angle of 60 ° is that the third angle has a degree measure of 30 °.
The area can be easily recognized by one of three formulas:
- through the height and side to which it descends;
- according to Heron's formula;
- on the sides and the corner between them.
The sides of a right-angled triangle, or rather the legs, converge at two heights. In order to find the third, it is necessary to consider the resulting triangle, and then, by the Pythagorean theorem, calculate the required length. In addition to this formula, there is also the ratio of the doubled area and the length of the hypotenuse. The most common expression among students is the former, as it requires less calculations.
Theorems applied to a right triangle
The geometry of a right triangle includes the use of theorems such as:
Online calculator.
Solving triangles.
The solution of a triangle is the finding of all its six elements (i.e. three sides and three angles) by any three given elements that define the triangle.
This math program finds side \\ (c \\), angles \\ (\\ alpha \\) and \\ (\\ beta \\) along user-specified sides \\ (a, b \\) and the angle between them \\ (\\ gamma \\)
The program not only gives the answer to the problem, but also displays the process of finding a solution.
This online calculator can be useful for senior students of secondary schools in preparation for tests and exams, when checking knowledge before the exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.
In this way, you can conduct your own teaching and / or teach your younger brothers or sisters, while the level of education in the field of the problems being solved increases.
If you are not familiar with the rules for entering numbers, we recommend that you familiarize yourself with them.
Number entry rules
Numbers can be set not only whole, but also fractional.
The whole and fractional parts in decimal fractions can be separated by either a point or a comma.
For example, you can enter decimal fractions as 2.5 or as 2.5
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A bit of theory.
Sine theorem
Theorem
The sides of the triangle are proportional to the sines of the opposite angles:
$$ \\ frac (a) (\\ sin A) \u003d \\ frac (b) (\\ sin B) \u003d \\ frac (c) (\\ sin C) $$
Cosine theorem
Theorem
Let in triangle ABC AB \u003d c, BC \u003d a, CA \u003d b. Then
The square of the side of a triangle is the sum of the squares of the other two sides minus twice the product of those sides multiplied by the cosine of the angle between them.
$$ a ^ 2 \u003d b ^ 2 + c ^ 2-2ba \\ cos A $$
Solving triangles
The solution of a triangle is the finding of all its six elements (i.e. three sides and three angles) by some three given elements that define the triangle.
Consider three problems for solving a triangle. In this case, we will use the following notation for the sides of the triangle ABC: AB \u003d c, BC \u003d a, CA \u003d b.
Solving a triangle on two sides and an angle between them
Given: \\ (a, b, \\ angle C \\). Find \\ (c, \\ angle A, \\ angle B \\)
Decision
1. By the cosine theorem, we find \\ (c \\):
$$ \\ cos A \u003d \\ frac (b ^ 2 + c ^ 2-a ^ 2) (2bc) $$
3. \\ (\\ angle B \u003d 180 ^ \\ circ - \\ angle A - \\ angle C \\)
Solving a triangle by a side and adjacent corners
Given: \\ (a, \\ angle B, \\ angle C \\). Find \\ (\\ angle A, b, c \\)
Decision
1. \\ (\\ angle A \u003d 180 ^ \\ circ - \\ angle B - \\ angle C \\)
$$ b \u003d a \\ frac (\\ sin B) (\\ sin A), \\ quad c \u003d a \\ frac (\\ sin C) (\\ sin A) $$
Solving a triangle on three sides
Given: \\ (a, b, c \\). Find \\ (\\ angle A, \\ angle B, \\ angle C \\)
Decision
1. By the cosine theorem, we obtain:
$$ \\ cos A \u003d \\ frac (b ^ 2 + c ^ 2-a ^ 2) (2bc) $$
2. Similarly, we find the angle B.
3. \\ (\\ angle C \u003d 180 ^ \\ circ - \\ angle A - \\ angle B \\)
Solving a triangle on two sides and an angle opposite a known side
Given: \\ (a, b, \\ angle A \\). Find \\ (c, \\ angle B, \\ angle C \\)
Decision
1. By the sine theorem we find \\ (\\ sin B \\) we obtain:
$$ \\ frac (a) (\\ sin A) \u003d \\ frac (b) (\\ sin B) \\ Rightarrow \\ sin B \u003d \\ frac (b) (a) \\ cdot \\ sin A $$
Let us introduce the notation: \\ (D \u003d \\ frac (b) (a) \\ cdot \\ sin A \\). Depending on the number D, cases are possible:
If D\u003e 1, such a triangle does not exist, since \\ (\\ sin B \\) cannot be greater than 1
If D \u003d 1, there is only one \\ (\\ angle B: \\ quad \\ sin B \u003d 1 \\ Rightarrow \\ angle B \u003d 90 ^ \\ circ \\)
If D If D 2. \\ (\\ angle C \u003d 180 ^ \\ circ - \\ angle A - \\ angle B \\)
3.Using the sine theorem, calculate the side c:
$$ c \u003d a \\ frac (\\ sin C) (\\ sin A) $$
A triangle is a primitive polygon bounded on the plane by three points and three line segments connecting these points in pairs. Angles in a triangle are sharp, obtuse, and straight. The sum of the angles in a triangle is continuous and equal to 180 degrees.
You will need
- Basic knowledge of geometry and trigonometry.
Instructions
1. We denote the lengths of the sides of the triangle a \u003d 2, b \u003d 3, c \u003d 4, and its angles u, v, w, each of which lies on the contrary on one side. According to the cosine theorem, the square of the length of the side of a triangle is equal to the sum of the squares of the lengths of the other 2 sides minus the double product of these sides by the cosine of the angle between them. That is, a ^ 2 \u003d b ^ 2 + c ^ 2 - 2bc * cos (u). Substitute in this expression the lengths of the sides and get: 4 \u003d 9 + 16 - 24cos (u).
2. Let us express from the obtained equality cos (u). We get the following: cos (u) \u003d 7/8. Next, we find the actual angle u. To do this, calculate arccos (7/8). That is, the angle u \u003d arccos (7/8).
3. Similarly, expressing the other sides in terms of the others, we find the remaining angles.
Note!
The value of one angle cannot exceed 180 degrees. The arccos () sign cannot contain a number greater than 1 and less than -1.
Useful advice
In order to find all three angles, it is not necessary to express all three sides, it is allowed to detect only 2 angles, and the third one can be obtained by subtracting the values \u200b\u200bof the remaining 2 from 180 degrees. This follows from the fact that the sum of all the angles of the triangle is a continuous value and is equal to 180 degrees.
A right-angled triangle is found in reality at almost every corner. Knowledge of the properties of a given figure, as well as the ability to calculate its area, will undoubtedly come in handy not only for solving problems in geometry, but also in life situations.
Triangle geometry
In elementary geometry, a right-angled triangle is a shape that consists of three connected line segments that form three corners (two acute and one straight line). The right-angled triangle is an original figure characterized by a number of important properties that form the foundation of trigonometry. Unlike a regular triangle, the sides of a rectangular shape have their own names:
- The hypotenuse is the longest side of the triangle, opposite the right angle.
- The legs are segments that form a right angle. Depending on the angle under consideration, the leg can be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). For non-rectangular triangles, the legs do not exist.
It is the ratio of legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right-angled triangle.
Rectangular triangle in reality
This figure has become widespread in reality. Triangles are used in design and engineering, so the calculation of the area of \u200b\u200bthe figure has to be done by engineers, architects and designers. The bases of tetrahedrons or prisms - three-dimensional figures that are easy to meet in everyday life - have the shape of a triangle. In addition, the gon is the simplest representation of a "flat" right-angled triangle in reality. The square is a locksmith, drawing, construction and carpentry tool that is used to draw corners by both schoolchildren and engineers.
Area of \u200b\u200ba triangle
The area of \u200b\u200ba geometric figure is a quantification of how much of the plane is bounded by the sides of the triangle. The area of \u200b\u200ban ordinary triangle can be found in five ways, using Heron's formula or by using variables such as the base, side, angle and radius of the inscribed or circumscribed circle in the calculations. The simplest area formula is:
where a is the side of the triangle, h is its height.
The formula for calculating the area of \u200b\u200ba right triangle is even simpler:
where a and b are legs.
Working with our online calculator, you can calculate the area of \u200b\u200ba triangle using three pairs of parameters:
- two legs;
- leg and adjacent angle;
- leg and opposite corner.
In tasks or everyday situations, you will be given different combinations of variables, so this form of the calculator allows you to calculate the area of \u200b\u200ba triangle in several ways. Let's look at a couple of examples.
Real life examples
Ceramic tile
Let's say you want to tiling the walls of your kitchen with ceramic tiles that are in the shape of a right-angled triangle. In order to determine the consumption of tiles, you must know the area of \u200b\u200bone cladding element and the total area of \u200b\u200bthe treated surface. Suppose you need to process 7 square meters. The length of the legs of one element is 19 cm each, then the area of \u200b\u200bthe tile will be equal to:
This means that the area of \u200b\u200bone element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that for finishing 7 square meters of a wall you will need 7 / 0.01805 \u003d 387 tiles.
School task
Suppose that in a school problem in geometry it is required to find the area of \u200b\u200ba right-angled triangle, knowing only that the side of one leg is 5 cm, and the value of the opposite angle is 30 degrees. Our online calculator comes with an illustration showing the sides and angles of a right triangle. If side a \u003d 5 cm, then its opposite angle is the alpha angle equal to 30 degrees. Enter this data into the calculator form and get the result:
Thus, the calculator not only calculates the area of \u200b\u200ba given triangle, but also determines the length of the adjacent leg and hypotenuse, as well as the value of the second angle.
Conclusion
Rectangular triangles are found in our lives literally on every corner. Determining the area of \u200b\u200bsuch figures will be useful to you not only when solving school geometry assignments, but also in everyday and professional activities.
