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. 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 private bank
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 transit and logistics. Liepaja port has all the chances to become the second Rotterdam in the foreseeable future. Rolands petersons private bank
Latvia as a distribution center for cargos from Asia and the Far East. Rolands petersons private bank
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 the 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 degree angle. A Pythagorean triangle is one whose sides are equal natural numbers; their lengths in this case are called the "Pythagorean triple".
egyptian triangle
In order for the current generation to learn geometry in the form in which it is taught at school now, it has been developed for several centuries. The fundamental point is the Pythagorean theorem. The sides of a rectangle are known to the whole 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". It is interesting that which is inscribed in the figure is equal to one. The name arose around the 5th century BC, when Greek philosophers traveled to Egypt.
When building the pyramids, architects and surveyors used the ratio 3:4:5. Such structures turned out to be proportional, pleasant to look at and spacious, and also rarely collapsed.
In order to build a right angle, the builders used a rope on which 12 knots were tied. In this case, the probability of constructing a right-angled triangle increased to 95%.
Signs of equality of figures
- 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 the equality of the 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 identical in the second criterion.
- When two figures are superimposed on each other, we rotate them in such a way that, when combined, they become one isosceles triangle. According to its property, the sides, or rather, the hypotenuses, are equal, as well as the angles at the base, which means that these figures are the same.
By the first sign, 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 according to the II sign, the essence of which is the equality of the leg and the acute angle.
Right angle triangle properties
Height lowered from 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 to the hypotenuse, is equal to half of it. 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 triangle, the properties of angles of 30 o, 45 o and 60 o 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 o, then the second sharp corner also 45 o. This suggests that the triangle is isosceles, and its legs are the same.
- The property of an angle of 60 degrees is that the third angle has a measure of 30 degrees.
The area is easy to find by one of three formulas:
- through the height and the side on which it descends;
- according to Heron's formula;
- along the sides and the angle between them.
The sides of a right triangle, or rather the legs, converge with two heights. In order to find the third, it is necessary to consider the resulting triangle, and then, using the Pythagorean theorem, calculate the required length. In addition to this formula, there is also the ratio of twice the area and the length of the hypotenuse. The most common expression among students is the first, as it requires less calculations.
Theorems that apply to a right triangle
The geometry of a right triangle includes the use of theorems such as:
Online calculator.
Solution of triangles.
The solution of a triangle is the finding of all its six elements (that is, three sides and three angles) by any three given elements that define the triangle.
This math program finds side \(c \), angles \(\alpha \) and \(\beta \) given 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 high school students general education schools in preparation for control work and exams, when testing knowledge before the exam, 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 training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.
If you are not familiar with the rules for entering numbers, we recommend that you familiarize yourself with them.
Rules for entering numbers
Numbers can be set not only whole, but also fractional.
The integer and fractional parts in decimal fractions can be separated by either a dot or a comma.
For example, you can enter decimals so 2.5 or so 2.5
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A bit of theory.
Sine theorem
Theorem
The sides of a triangle are proportional to the sines of the opposite angles:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) = \frac(c)(\sin C) $$
Cosine theorem
Theorem
Let in triangle ABC AB = c, BC = a, CA = b. Then
The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them.
$$ a^2 = b^2+c^2-2ba \cos A $$
Solving Triangles
The solution of a triangle is the finding of all its six elements (i.e. three parties and three angles) according to 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 = c, BC = a, CA = b.
Solution of a triangle given two sides and an angle between them
Given: \(a, b, \angle C \). Find \(c, \angle A, \angle B \)
Solution
1. By the law of cosines we find \(c\):
$$ \cos A = \frac( b^2+c^2-a^2 )(2bc) $$
3. \(\angle B = 180^\circ -\angle A -\angle C \)
Solution of a triangle given a side and adjacent angles
Given: \(a, \angle B, \angle C \). Find \(\angle A, b, c \)
Solution
1. \(\angle A = 180^\circ -\angle B -\angle C \)
$$ b = a \frac(\sin B)(\sin A), \quad c = a \frac(\sin C)(\sin A) $$
Solving a Triangle with Three Sides
Given: \(a, b, c\). Find \(\angle A, \angle B, \angle C \)
Solution
1. According to the cosine theorem, we get:
$$ \cos A = \frac(b^2+c^2-a^2)(2bc) $$
2. Similarly, we find the angle B.
3. \(\angle C = 180^\circ -\angle A -\angle B \)
Solving a triangle given two sides and an angle opposite a known side
Given: \(a, b, \angle A\). Find \(c, \angle B, \angle C \)
Solution
1. By the sine theorem we find \(\sin B \) we get:
$$ \frac(a)(\sin A) = \frac(b)(\sin B) \Rightarrow \sin B = \frac(b)(a) \cdot \sin A $$
Let's introduce the notation: \(D = \frac(b)(a) \cdot \sin A \). Depending on the number D, the following cases are possible:
If D > 1, such a triangle does not exist, because \(\sin B \) cannot be greater than 1
If D = 1, there is a unique \(\angle B: \quad \sin B = 1 \Rightarrow \angle B = 90^\circ \)
If D If D 2. \(\angle C = 180^\circ -\angle A -\angle B \)
3. Using the sine theorem, we calculate the side c:
$$ c = a \frac(\sin C)(\sin A) $$
A triangle is a primitive polygon bounded on a plane by three points and three line segments connecting these points in pairs. The angles in a triangle are acute, obtuse and right. The sum of the angles in a triangle is continuous and equals 180 degrees.
You will need
- Basic knowledge in geometry and trigonometry.
Instruction
1. Let us denote the lengths of the sides of the triangle a=2, b=3, c=4, and its angles u, v, w, each of which lies on the opposite side of one side. According to the law of cosines, the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of 2 other sides minus twice the product of these sides by the cosine of the angle between them. That is, a^2 = b^2 + c^2 - 2bc*cos(u). We substitute the lengths of the sides into this expression and get: 4 \u003d 9 + 16 - 24cos (u).
2. Let us express cos(u) from the obtained equality. We get the following: cos(u) = 7/8. Next, we find the actual angle u. To do this, we calculate arccos(7/8). That is, the angle u = arccos(7/8).
3. Similarly, expressing the other sides in terms of the rest, we find the remaining angles.
Note!
The value of one angle cannot exceed 180 degrees. The arccos() sign cannot contain a number larger than 1 and smaller than -1.
Useful advice
In order to detect all three angles, it is not necessary to express all three sides, it is allowed to detect only 2 angles, and the 3rd one can be obtained by subtracting the values of the remaining 2 from 180 degrees. This follows from the fact that the sum of all the angles of a triangle is continuous and equals 180 degrees.
A right triangle is found in reality on almost every corner. Knowledge of the properties of this figure, as well as the ability to calculate its area, will undoubtedly be useful to you not only for solving problems in geometry, but also in life situations.
triangle geometry
In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). Right triangle - original figure, characterized by a number of important properties that form the foundation of trigonometry. Unlike an ordinary triangle, the sides of a rectangular figure have their own names:
- The hypotenuse is the longest side of a triangle that lies opposite the right angle.
- Legs - segments that form a right angle. Depending on the angle under consideration, the leg may be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). There are no legs for non-rectangular triangles.
It is the ratio of the legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right triangle.
Right triangle in reality
This figure is widely used in reality. Triangles are used in design and technology, so the calculation of the area of \u200b\u200bthe figure has to be done by engineers, architects and designers. The bases of tetrahedra or prisms have the shape of a triangle - three-dimensional figures that are easy to meet in everyday life. In addition, a square is the simplest representation of a "flat" right triangle in reality. A square is a locksmith, drawing, construction and carpentry tool that is used to build corners by both schoolchildren and engineers.
Area of a triangle
Square geometric figure is a quantitative estimate of how much of the plane is bounded by the sides of the triangle. The area of an ordinary triangle can be found in five ways, using Heron's formula or operating in calculations with such variables as the base, side, angle and radius of the inscribed or circumscribed circle. The most simple formula area is expressed as:
where a is the side of the triangle, h is its height.
The formula for calculating the area of a right triangle is even simpler:
where a and b are legs.
Working with our online calculator, you can calculate the area of a triangle using three pairs of parameters:
- two legs;
- leg and adjacent angle;
- leg and opposite angle.
In tasks or everyday situations, you will be given different combinations variables, so this form of calculator allows you to calculate the area of a triangle in several ways. Let's look at a couple of examples.
Real life examples
Ceramic tile
Let's say you want to clad the walls of the kitchen ceramic tiles, which has the shape of a right triangle. In order to determine the consumption of tiles, you must find out the area of \u200b\u200bone element of the cladding and the total area of \u200b\u200bthe surface to be treated. Let 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 one element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that to finish 7 square meters of a wall, you will need 7 / 0.01805 = 387 facing tiles.
school task
Suppose that in a school geometry problem it is required to find the area of a right 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 is accompanied by an illustration showing the sides and angles of a right triangle. If side a = 5 cm, then its opposite angle is the angle alpha, equal to 30 degrees. Enter this data into the calculator form and get the result:
Thus, the calculator not only calculates the area of a given triangle, but also determines the length 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 such figures will be useful to you not only when solving school assignments in geometry, but also in everyday and professional activities.
