A triangle (from the point of view of Euclid's space) is such a geometric figure, which is formed by three segments connecting three points that do not lie on one straight line. The three points that form a triangle are called its vertices, and the line segments connecting the vertices are called sides of the triangle. What are triangles?
Equal Triangles
There are three signs of the equality of triangles. What triangles are called equal? These are the ones who:
- two sides and the angle between these sides are equal;
- one side and two angles adjacent to it are equal;
- all three sides are equal.
Right triangles have the following signs of equality:
- along an acute angle and hypotenuse;
- along an acute angle and leg;
- on two legs;
- along the hypotenuse and cathetus.
What are triangles
According to the number of equal sides, a triangle can be:
- Equilateral. It is a triangle with three equal sides. All angles in an equilateral triangle are 60 degrees. In addition, the centers of the circumscribed and inscribed circles coincide.
- Unequilateral. A triangle with no equal sides.
- Isosceles. It is a triangle with two equal sides. Two identical sides are the sides, and the third side is the base. In such a triangle, the bisector, median and height coincide if they are lowered to the base.
According to the size of the angles, a triangle can be:
- Obtuse - when one of the angles has a value of more than 90 degrees, that is, when it is obtuse.
- Acute-angled - if all three angles in the triangle are acute, that is, they have a value of less than 90 degrees.
- Which triangle is called a right triangle? This is one that has one right angle equal to 90 degrees. The legs in it will be called the two sides that form this angle, and the hypotenuse is the side opposite the right angle.
Basic properties of triangles
- A smaller angle always lies opposite the smaller side, and a larger angle always lies opposite the larger side.
- Equal angles always lie opposite equal sides, and opposite sides always lie different angles. In particular, in an equilateral triangle, all angles have the same value.
- In any triangle, the sum of the angles is 180 degrees.
- An external angle can be obtained by extending one of its sides to a triangle. The value of the outer angle will be equal to the sum of the inner angles not adjacent to it.
- The side of a triangle is greater than the difference of its other two sides, but less than their sum.
In the spatial geometry of Lobachevsky, the sum of the angles of a triangle will always be less than 180 degrees. On a sphere, this value is greater than 180 degrees. The difference between 180 degrees and the sum of the angles of a triangle is called a defect.
A triangle in which all sides are not the same length is called versatile.
A triangle with two equal sides is denoted as isosceles. The same sides are called lateral, the third party basis. The following definition would be equally true bases of a triangle is the side of an isosceles triangle that is not equal to the other two sides.
IN isosceles triangle base angles are equal. Height, median, bisector isosceles triangle, drawn to its base, are combined.
Triangle, with all sides the same, is denoted as equilateral or correct. In an equilateral triangle, all angles are 60°, and the centers of the inscribed and circumscribed circles are aligned.
Types of triangles depending on the parameters of the angles.
A triangle in which only angles less than 90 0 (acute) are called acute-angled.
A triangle in which an angle of 90 0 is represented is called rectangular. The sides of a triangle forming a right angle are usually denoted legs, and the side opposite the right angle - hypotenuse.
The science of geometry tells us what a triangle, square, cube is. In the modern world, it is studied in schools by everyone without exception. Also, a science that directly studies what a triangle is and what properties it has is trigonometry. She explores in detail all the phenomena associated with data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems related to them.
What is a triangle? Definition
This is a flat polygon. It has three corners, which is clear from its name. It also has three sides and three vertices, the first of which are segments, the second are points. Knowing what two angles are equal to, you can find the third one by subtracting the sum of the first two from the number 180.
What are triangles?
They can be classified according to various criteria.
First of all, they are divided into acute-angled, obtuse-angled and rectangular. The first have acute angles, that is, those that are less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. Acute triangles also include equilateral triangles. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.
Right triangle
It is impossible not to talk about what a right triangle is.
Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The other two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. With its help, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can recall the isosceles. This is one in which two of the sides are equal, and two of the angles are also equal.
What is the leg and hypotenuse?
The leg is one of the sides of a triangle that form an angle of 90 degrees. The hypotenuse is the remaining side that is opposite the right angle. From it, a perpendicular can be lowered onto the leg. The ratio of the adjacent leg to the hypotenuse is called the cosine, and the opposite is called the sine.
- what are its features?
It is rectangular. Its legs are three and four, and the hypotenuse is five. If you saw that the legs of this triangle are equal to three and four, you can be sure that the hypotenuse will be equal to five. Also, according to this principle, it can be easily determined that the leg will be equal to three if the second is equal to four, and the hypotenuse is five. To prove this statement, you can apply the Pythagorean theorem. If two legs are 3 and 4, then 9 + 16 \u003d 25, the root of 25 is 5, that is, the hypotenuse is 5. Also, the Egyptian triangle is called a right triangle, whose sides are 6, 8 and 10; 9, 12 and 15 and other numbers with a ratio of 3:4:5.
What else could be a triangle?
Triangles can also be inscribed and circumscribed. The figure around which the circle is described is called inscribed, all its vertices are points lying on the circle. A circumscribed triangle is one in which a circle is inscribed. All its sides are in contact with it at certain points.
How is
The area of any figure is measured in square units (square meters, square millimeters, square centimeters, square decimeters, etc.). This value can be calculated in a variety of ways, depending on the type of triangle. The area of any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from the opposite angle, and dividing this figure by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle between these sides, and divide this by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this, you need to find half the perimeter. Then alternately subtract different sides from this number and multiply the four values obtained. Next, find out the number that came out. The area of an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by which is circumscribed around it times four.
The area of the described triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: we square the side, multiply the resulting figure by the root of three, then divide this number by four. Similarly, you can calculate the height of a triangle in which all sides are equal, for this you need to multiply one of them by the root of three, and then divide this number by two.
Triangle theorems
The main theorems that are associated with this figure are the Pythagorean theorem, described above, and cosines. The second (sine) is that if you divide any side by the sine of the angle opposite to it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosine) is that if the sum of the squares of the two sides is subtracted from their product, multiplied by two and the cosine of the angle located between them, then the square of the third side will be obtained.
Dali triangle - what is it?
Many, faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. The Dali Triangle is the common name for three places that are closely associated with the life of the famous artist. Its “tops” are the house where Salvador Dali lived, the castle that he gave to his wife, and the museum of surrealistic paintings. During a tour of these places, you can learn many interesting facts about this original creative artist, known throughout the world.
