A triangle (from the point of view of Euclidean space) is such a geometric figure, which is formed by three segments connecting three points that do not lie on the same straight line. The three points that formed the triangle are called its vertices, and the segments connecting the vertices are called the sides of the triangle. What types of triangles are there?
Equal triangles
There are three signs that triangles are equal. Which triangles are called equal? These are those who:
- two sides and the angle between these sides are equal;
- one side and two adjacent angles are equal;
- all three sides are equal.
Right triangles have the following signs of equality:
- along the acute angle and hypotenuse;
- along an acute angle and leg;
- on two legs;
- along the hypotenuse and leg.
What types of triangles are there?
Based on the number of equal sides, a triangle can be:
- Equilateral. This is a triangle with three equal sides. All angles in an equilateral triangle are equal to 60 degrees. In addition, the centers of the circumscribed and inscribed circles coincide.
- Unequilateral. A triangle that has no equal sides.
- Isosceles. This 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 altitude 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 is more than 90 degrees, that is, when it is obtuse.
- Acute - if all three angles in the triangle are acute, that is, they measure less than 90 degrees.
- Which triangle is called a right triangle? This is one that has one right angle equal to 90 degrees. The two sides that form this angle will be called the legs, and the hypotenuse will be the side opposite the right angle.
Basic properties of triangles
- The smaller angle always lies opposite the smaller side, and the larger angle always lies opposite the larger side.
- Equal angles always lie opposite equal sides, and always lie opposite different sides 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 the sides of a triangle. The size of the external angle will be equal to the sum of non-adjacent internal corners.
- A side of a triangle is greater than the difference of its two other sides, but less than their sum.
In Lobachevsky's spatial geometry, the sum of the angles of a triangle will always be less than 180 degrees. On a sphere this value is more than 180 degrees. The difference between 180 degrees and the sum of the angles of the triangle is called a defect.
When studying mathematics, students begin to become familiar with different types of geometric shapes. Today we will talk about different types of triangles.
Definition
Geometric figures that consist of three points that are not on the same line are called triangles.
The segments connecting the points are called sides, and the points are called vertices. Vertices are designated in capital letters, for example: A, B, C.
The sides are designated by the names of the two points from which they consist - AB, BC, AC. Intersecting, the sides form angles. Down side is considered the base of the figure.
Rice. 1. Triangle ABC.
Types of triangles
Triangles are classified by angles and sides. Each type of triangle has its own properties.
There are three types of triangles at the corners:
- acute-angled;
- rectangular;
- obtuse-angled.
All angles acute-angled triangles are acute, that is, the degree measure of each is no more than 90 0.
Rectangular a triangle contains a right angle. The other two angles will always be acute, since otherwise the sum of the angles of the triangle will exceed 180 degrees, and this is impossible. The side that is opposite right angle, is called the hypotenuse, and the other two legs. The hypotenuse is always larger than the leg.
Obtuse the triangle contains an obtuse angle. That is, an angle greater than 90 degrees. The other two angles in such a triangle will be acute.
Rice. 2. Types of triangles at the corners.
A Pythagorean triangle is a rectangle whose sides are 3, 4, 5.
Moreover, the larger side is the hypotenuse.
Such triangles are often used to make simple tasks in geometry. Therefore, remember: if two sides of a triangle are equal to 3, then the third will definitely be 5. This will simplify the calculations.
Types of triangles on the sides:
- equilateral;
- isosceles;
- versatile.
Equilateral a triangle is a triangle in which all sides are equal. All angles of such a triangle are equal to 60 0, that is, it is always acute.
Isosceles triangle - a triangle with only two sides equal. These sides are called lateral, and the third is called the base. In addition, the angles at the base of an isosceles triangle are equal and always acute.
Versatile or an arbitrary triangle is a triangle in which all lengths and all angles are not equal to each other.
If the problem does not contain any clarifications about the figure, then it is generally accepted that we are talking about an arbitrary triangle.
Rice. 3. Types of triangles on the sides.
The sum of all angles of a triangle, regardless of its type, is 1800.
Opposite the larger angle is the larger side. And also the length of any side is always less than the sum of its other two sides. These properties are confirmed by the triangle inequality theorem.
There is a concept of the golden triangle. This is an isosceles triangle in which two sides are proportional to the base and equal a certain number. In such a figure, the angles are proportional to the ratio 2:2:1.
Task:
Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?
Solution:
To solve this task you need to use the inequality a
What have we learned?
From of this material From the 5th grade mathematics course, we learned that triangles are classified according to their sides and the size of their angles. Triangles have certain properties that can be used to solve problems.
The science of geometry tells us what a triangle, square, and cube are. IN modern world it is studied in schools by everyone without exception. Also, the science that studies directly what a triangle is and what properties it has is trigonometry. She explores in detail all phenomena related to data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems associated with them.
What is a triangle? Definition
This is a flat polygon. It has three corners, as is clear from its name. It also has three sides and three vertices, the first of them are segments, the second are points. Knowing what two angles are equal to, you can find the third by subtracting the sum of the first two from the number 180.
What types of triangles are there?
They can be classified according to various criteria.
First of all, they are divided into acute-angled, obtuse-angled and rectangular. The former have acute angles, that is, those that are equal to 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 remaining two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. Using it, 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 also recall an isosceles triangle. This is one in which two of the sides are equal, and two angles are also equal.
What are leg and hypotenuse?
A leg is one of the sides of a triangle that forms an angle of 90 degrees. The hypotenuse is the remaining side that is opposite the right angle. You can lower a perpendicular from it onto the leg. The ratio of the adjacent side to the hypotenuse is called cosine, and the opposite side is called sine.
- what are its features?
It's rectangular. Its legs are three and four, and its hypotenuse is five. If you see that the legs of a given triangle are equal to three and four, you can rest assured that the hypotenuse will be equal to five. Also, using this principle, you can easily determine that the leg will be equal to three if the second is equal to four, and the hypotenuse is equal to five. To prove this statement, you can apply the Pythagorean theorem. If two legs are equal to 3 and 4, then 9 + 16 = 25, the root of 25 is 5, that is, the hypotenuse is equal to 5. An Egyptian triangle is also a right triangle whose sides are equal to 6, 8 and 10; 9, 12 and 15 and other numbers with the ratio 3:4:5.
What else could a triangle be?
Triangles can also be inscribed or 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 into which a circle is inscribed. All its sides come into contact with it at certain points.
How is it located?
The area of any figure is measured in square units (sq. meters, sq. millimeters, sq. centimeters, sq. 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 corner, 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 located between these sides, and divide this result 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 resulting four values. Next, find from 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 that circumscribed around it, multiplied by four.
The area of a circumscribed 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: square the side, multiply the resulting figure by the root of three, then divide this number by four. In a similar way, you can calculate the height of a triangle in which all sides are equal; to do this, you need to multiply one of them by the root of three, and then divide this number by two.
Theorems related to triangle
The main theorems that are associated with this figure are the Pythagorean theorem described above and cosines. The second (of sines) is that if you divide any side by the sine of the angle opposite it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosines) is that if from the sum of the squares of the two sides we subtract their product, multiplied by two and the cosine of the angle located between them, then we get the square of the third side.
Dali Triangle - what is it?
Many, when 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 connected with the life of the famous artist. Its “peaks” are the house in which Salvador Dali lived, the castle that he gave to his wife, as well as the museum of surrealist paintings. You can learn a lot during a tour of these places. interesting facts about this unique creative artist, known all over the world.
Triangle - definition and general concepts
A triangle is a simple polygon consisting of three sides and having the same number of angles. Its planes are limited by 3 points and 3 segments connecting these points in pairs.
All vertices of any triangle, regardless of its variety, are designated by capital Latin letters, and its sides are depicted with the corresponding designations of opposite vertices, but not in capital letters, but small. So, for example, a triangle with vertices labeled A, B and C has sides a, b, c.
If we consider a triangle in Euclidean space, then it is a geometric figure that is formed using three segments connecting three points that do not lie on the same straight line.
Look carefully at the picture shown above. On it, points A, B and C are the vertices of this triangle, and its segments are called the sides of the triangle. Each vertex of this polygon forms angles inside it.
Types of triangles
According to the size of the angles of triangles, they are divided into such varieties as: Rectangular;
Acute angular;
Obtuse.
Rectangular triangles include those that have one right angle and the other two have acute angles.
Acute triangles are those in which all its angles are acute.
And if a triangle has one obtuse angle and the other two acute angles, then such a triangle is classified as obtuse.
Each of you understands perfectly well that not all triangles have equal sides. And according to the length of its sides, triangles can be divided into:
Isosceles;
Equilateral;
Versatile.
Task: Draw different types triangles. Define them. What difference do you see between them?
Basic properties of triangles
Although these simple polygons may differ from each other in the size of their angles or sides, each triangle has the basic properties that are characteristic of this figure.
In any triangle:
The total sum of all its angles is 180º.
If it belongs to equilaterals, then each of its angles is 60º.
An equilateral triangle has equal and equal angles.
The smaller the side of the polygon, the smaller the angle opposite it, and vice versa, the larger angle is opposite the larger side.
If the sides are equal, then opposite them are equal angles, and vice versa.
If we take a triangle and extend its side, we end up with external corner. It is equal to the sum of the internal angles.
In any triangle, its side, no matter which one you choose, will still be less than the sum of the other 2 sides, but more than their difference:
1. a< b + c, a >b–c;
2. b< a + c, b >a–c;
3. c< a + b, c >a–b.
Exercise
The table shows the already known two angles of the triangle. Knowing total amount of all angles, find what the third angle of the triangle is equal to and put it in the table:
1. How many degrees does the third angle have?
2. What type of triangle does it belong to?
Tests for equivalence of triangles
I sign
II sign
III sign
Height, bisector and median of a triangle
The altitude of a triangle - the perpendicular drawn from the vertex of the figure to its opposite side is called the altitude of the triangle. All altitudes of a triangle intersect at one point. The point of intersection of all 3 altitudes of a triangle is its orthocenter.
A segment drawn from a given vertex and connecting it at the middle of the opposite side is the median. Medians, as well as altitudes of a triangle, have one common point of intersection, the so-called center of gravity of the triangle or centroid.
The bisector of a triangle is a segment connecting the vertex of an angle and a point on the opposite side, and also dividing this angle in half. All bisectors of a triangle intersect at one point, which is called the center of the circle inscribed in the triangle.
The segment that connects the midpoints of 2 sides of a triangle is called the midline.
Historical reference
A figure such as a triangle was known back in Ancient times. This figure and its properties were mentioned on Egyptian papyri four thousand years ago. A little later, thanks to the Pythagorean theorem and Heron’s formula, the study of the properties of a triangle moved to more high level, but still, this happened more than two thousand years ago.
In XV – 16th centuries They began to conduct a lot of research on the properties of a triangle, and as a result such a science as planimetry arose, which was called “New Triangle Geometry”.
Russian scientist N.I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application in mathematics, physics and cybernetics.
Thanks to knowledge of the properties of triangles, such a science as trigonometry arose. It turned out to be necessary for a person in his practical needs, since its use is simply necessary when drawing up maps, measuring areas, and even when designing various mechanisms.
What is the most famous triangle you know? This is of course the Bermuda Triangle! It received this name in the 50s because of the geographical location of the points (vertices of the triangle), within which, according to the existing theory, anomalies associated with it arose. The vertices of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.
Assignment: What theories about Bermuda Triangle did you hear?
Did you know that in Lobachevsky’s theory, when adding the angles of a triangle, their sum always has a result less than 180º. In Riemann's geometry, the sum of all the angles of a triangle is greater than 180º, and in the works of Euclid it is equal to 180 degrees.
Homework
Solve a crossword puzzle on a given topic
Questions for the crossword:
1. What is the name of the perpendicular that is drawn from the vertex of the triangle to the straight line located on the opposite side?
2. How, in one word, can you call the sum of the lengths of the sides of a triangle?
3. Name a triangle whose two sides are equal?
4. Name a triangle that has an angle equal to 90°?
5. What is the name of the largest side of the triangle?
6. What is the name of the side of an isosceles triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle in which one of the angles exceeds 90°?
9. The name of the segment connecting the top of our figure with the middle of the opposite side?
10. In a simple polygon ABC, the capital letter A is...?
11. What is the name of the segment dividing the angle of a triangle in half?
Questions on the topic of triangles:
1. Define it.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is its sum of angles?
5. What types of this simple polygon do you know?
6. Name the points of the triangles that are called remarkable.
7. What device can you use to measure the angle?
8. If the clock hands show 21 o'clock. What angle do the hour hands make?
9. At what angle does a person turn if he is given the command “left”, “circle”?
10. What other definitions do you know that are associated with a figure that has three angles and three sides?
Perhaps the most basic, simple and interesting figure in geometry is the triangle. I know high school its basic properties are studied, but sometimes knowledge on this topic is incomplete. The types of triangles initially determine their properties. But this view remains mixed. Therefore, now let’s look at this topic in a little more detail.
The types of triangles depend on the degree measure of the angles. These figures are acute, rectangular and obtuse. If all angles do not exceed 90 degrees, then the figure can safely be called acute. If at least one angle of the triangle is 90 degrees, then you are dealing with a rectangular subspecies. Accordingly, in all other cases the one under consideration is called obtuse-angled.
There are many problems for acute-angled subtypes. Distinctive feature is the internal location of the intersection points of bisectors, medians and altitudes. In other cases, this condition may not be met. It is not difficult to determine the type of triangle figure. It is enough to know, for example, the cosine of each angle. If any values less than zero, which means that the triangle is obtuse in any case. In the case of a zero indicator, the figure has a right angle. All positive values are guaranteed to tell you that you are looking at an angular view.
One cannot help but mention the regular triangle. This is the most perfect view, where all intersection points of medians, bisectors and heights coincide. The center of the inscribed and circumscribed circle also lies in the same place. To solve problems, you need to know only one side, since the angles are initially given to you, and the other two sides are known. That is, the figure is specified by only one parameter. They exist main feature- equality of two sides and angles at the base.
Sometimes the question arises as to whether a triangle with given sides exists. What you are really asking is whether the given description fits the main species. For example, if the sum of two sides is less than the third, then in reality such a figure does not exist at all. If the task asks you to find the cosines of the angles of a triangle with sides of 3,5,9, then the obvious can be explained without complex mathematical techniques. Suppose you want to get from point A to point B. The distance in a straight line is 9 kilometers. However, you remembered that you need to go to point C in the store. The distance from A to C is 3 kilometers, and from C to B is 5. Thus, it turns out that when moving through the store, you will walk one kilometer less. But since point C is not located on straight AB, you will have to walk an extra distance. There is a contradiction here. This is, of course, a conditional explanation. Mathematics knows more than one way to prove that all types of triangles obey the basic identity. It states that the sum of two sides is greater than the length of the third.
Any type has the following properties:
1) The sum of all angles is 180 degrees.
2) There is always an orthocenter - the point of intersection of all three heights.
3) All three medians drawn from the vertices of the interior angles intersect in one place.
4) A circle can be drawn around any triangle. You can also inscribe a circle so that it has only three points of contact and does not extend beyond the outer sides.
Now you are familiar with the main properties that they have different kinds triangles. In the future, it is important to understand what you are dealing with when solving a problem.