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A triangle has all sides equal. Properties of a triangle

The simplest polygon that is studied in school is a triangle. It is more understandable for students and encounters fewer difficulties. Despite the fact that there are different types of triangles, which have special properties.

What shape is called a triangle?

Formed by three points and segments. The first ones are called vertices, the second ones are called sides. Moreover, all three segments must be connected so that angles are formed between them. Hence the name of the “triangle” figure.

Differences in names across corners

Since they can be acute, obtuse and straight, the types of triangles are determined by these names. Accordingly, there are three groups of such figures.

First. If all the angles of a triangle are acute, then it will be called acute. Everything is logical.

Second. One of the angles is obtuse, which means the triangle is obtuse. It couldn't be simpler.

Third. There is an angle equal to 90 degrees, which is called a right angle. The triangle becomes rectangular.

Differences in names on the sides

Depending on the characteristics of the sides, the following types of triangles are distinguished:

the general case is scalene, in which all sides are of arbitrary length;

isosceles, two sides of which have the same numerical values;

equilateral, the lengths of all its sides are the same.

If the problem does not specify a specific type of triangle, then you need to draw an arbitrary one. Which all the corners are sharp and the sides have different lengths.

Properties common to all triangles

If you add up all the angles of a triangle, you get a number equal to 180º. And it doesn't matter what type it is. This rule always applies.
The numerical value of any side of a triangle is less than the other two added together. Moreover, it is greater than their difference.
Every external corner has the value that is obtained by adding two internal ones that are not adjacent to it. Moreover, it is always larger than the internal one adjacent to it.
The smallest angle is always opposite the smaller side of the triangle. And vice versa, if the side is large, then the angle will be the largest.

These properties are always valid, no matter what types of triangles are considered in the problems. All the rest follow from specific features.

Properties of an isosceles triangle

The angles that are adjacent to the base are equal.
The height, which is drawn to the base, is also the median and bisector.
The altitudes, medians and bisectors, which are built to the lateral sides of the triangle, are respectively equal to each other.

Properties of an equilateral triangle

If there is such a figure, then all the properties described a little above will be true. Because an equilateral will always be isosceles. But not vice versa; an isosceles triangle will not necessarily be equilateral.

All its angles are equal to each other and have a value of 60º.
Any median of an equilateral triangle is its altitude and bisector. Moreover, they are all equal to each other. To determine their values, there is a formula that consists of the product of the side and the square root of 3 divided by 2.

Properties of a right triangle

Two acute angles add up to 90º.
The length of the hypotenuse is always greater than that of any of the legs.
The numerical value of the median drawn to the hypotenuse is equal to its half.
The leg is equal to the same value if it lies opposite an angle of 30º.
The height, which is drawn from the vertex with a value of 90º, has a certain mathematical dependence on the legs: 1/n 2 = 1/a 2 + 1/b 2. Here: a, b - legs, n - height.

Problems with different types of triangles

No. 1. Given an isosceles triangle. Its perimeter is known and equal to 90 cm. We need to find out its sides. As an additional condition: the side side is 1.2 times smaller than the base.

The value of the perimeter directly depends on the quantities that need to be found. The sum of all three sides will give 90 cm. Now you need to remember the sign of a triangle, according to which it is isosceles. That is, the two sides are equal. You can create an equation with two unknowns: 2a + b = 90. Here a is the side, b is the base.

Now it's time for an additional condition. Following it, the second equation is obtained: b = 1.2a. You can substitute this expression into the first one. It turns out: 2a + 1.2a = 90. After transformations: 3.2a = 90. Hence a = 28.125 (cm). Now it is easy to find out the basis. This is best done from the second condition: b = 1.2 * 28.125 = 33.75 (cm).

To check, you can add three values: 28.125 * 2 + 33.75 = 90 (cm). That's right.

Answer: The sides of the triangle are 28.125 cm, 28.125 cm, 33.75 cm.

No. 2. The side of an equilateral triangle is 12 cm. You need to calculate its height.

Solution. To find the answer, it is enough to return to the moment where the properties of the triangle were described. This is the formula for finding the height, median and bisector of an equilateral triangle.

n = a * √3 / 2, where n is the height and a is the side.

Substitution and calculation give the following result: n = 6 √3 (cm).

There is no need to memorize this formula. It is enough to remember that the height divides the triangle into two rectangular ones. Moreover, it turns out to be a leg, and the hypotenuse in it is the side of the original one, the second leg is half of the known side. Now you need to write down the Pythagorean theorem and derive a formula for height.

Answer: height is 6 √3 cm.

No. 3. Given MKR is a triangle, in which angle K makes 90 degrees. The sides MR and KR are known, they are equal to 30 and 15 cm, respectively. We need to find out the value of angle P.

Solution. If you make a drawing, it becomes clear that MR is the hypotenuse. Moreover, it is twice as large as the side of the KR. Again you need to turn to the properties. One of them has to do with angles. From it it is clear that the KMR angle is 30º. This means that the desired angle P will be equal to 60º. This follows from another property, which states that the sum of two sharp corners should be 90º.

Answer: angle P is 60º.

No. 4. We need to find all the angles of an isosceles triangle. It is known about it that the external angle from the angle at the base is 110º.

Solution. Since only the external angle is given, this is what you need to use. It forms an unfolded angle with the internal one. This means that in total they will give 180º. That is, the angle at the base of the triangle will be equal to 70º. Since it is isosceles, the second angle has the same value. It remains to calculate the third angle. According to a property common to all triangles, the sum of the angles is 180º. This means that the third will be defined as 180º - 70º - 70º = 40º.

Answer: the angles are 70º, 70º, 40º.

No. 5. It is known that in an isosceles triangle the angle opposite the base is 90º. There is a point marked on the base. The segment connecting it to a right angle divides it in the ratio of 1 to 4. You need to find out all the angles of the smaller triangle.

Solution. One of the angles can be determined immediately. Since the triangle is right-angled and isosceles, those that lie at its base will be 45º each, that is, 90º/2.

The second of them will help you find the relation known in the condition. Since it is equal to 1 to 4, the parts into which it is divided are only 5. This means that to find out the smaller angle of a triangle you need 90º/5 = 18º. It remains to find out the third. To do this, you need to subtract 45º and 18º from 180º (the sum of all angles of the triangle). The calculations are simple, and you get: 117º.

Today we are going to the country of Geometry, where we will get acquainted with various types triangles.

Consider geometric shapes and find the “extra” one among them (Fig. 1).

Rice. 1. Illustration for example

We see that figures No. 1, 2, 3, 5 are quadrilaterals. Each of them has its own name (Fig. 2).

Rice. 2. Quadrilaterals

This means that the “extra” figure is a triangle (Fig. 3).

Rice. 3. Illustration for example

A triangle is a figure that consists of three points that do not lie on the same line and three segments connecting these points in pairs.

The points are called vertices of the triangle, segments - his parties. The sides of the triangle form There are three angles at the vertices of a triangle.

The main features of a triangle are three sides and three corners. According to the size of the angle, triangles are acute, rectangular and obtuse.

A triangle is called acute-angled if all three of its angles are acute, that is, less than 90° (Fig. 4).

Rice. 4. Acute triangle

A triangle is called rectangular if one of its angles is 90° (Fig. 5).

Rice. 5. Right Triangle

A triangle is called obtuse if one of its angles is obtuse, that is, more than 90° (Fig. 6).

Rice. 6. Obtuse triangle

Based on the number of equal sides, triangles are equilateral, isosceles, scalene.

An isosceles triangle is one in which two sides are equal (Fig. 7).

Rice. 7. Isosceles triangle

These sides are called lateral, third party - basis. In an isosceles triangle, the base angles are equal.

There are isosceles triangles acute and obtuse(Fig. 8) .

Rice. 8. Acute and obtuse isosceles triangles

An equilateral triangle is one in which all three sides are equal (Fig. 9).

Rice. 9. Equilateral triangle

In an equilateral triangle all angles are equal. Equilateral triangles Always acute-angled.

A scalene triangle is one in which all three sides have different lengths (Fig. 10).

Rice. 10. Scalene triangle

Complete the task. Distribute these triangles into three groups (Fig. 11).

Rice. 11. Illustration for the task

First, let's distribute according to the size of the angles.

Acute triangles: No. 1, No. 3.

Right triangles: No. 2, No. 6.

Obtuse triangles: No. 4, No. 5.

We will distribute the same triangles into groups according to the number of equal sides.

Scalene triangles: No. 4, No. 6.

Isosceles triangles: No. 2, No. 3, No. 5.

Equilateral triangle: No. 1.

Look at the pictures.

Think about what piece of wire each triangle was made from (Fig. 12).

Rice. 12. Illustration for the task

You can think like this.

The first piece of wire is divided into three equal parts, so you can make an equilateral triangle from it. He is shown third in the picture.

The second piece of wire is divided into three different parts, so it can be used to make a scalene triangle. It is shown first in the picture.

The third piece of wire is divided into three parts, where two parts have the same length, which means that an isosceles triangle can be made from it. In the picture he is shown second.

Today in class we learned about different types of triangles.

References

M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
M.I. Moro. Math lessons: Methodical recommendations for the teacher. 3rd grade. - M.: Education, 2012.
Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
"School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
S.I. Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.

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Homework

1. Complete the phrases.

a) A triangle is a figure that consists of ... that do not lie on the same line, and ... that connect these points in pairs.

b) The points are called … , segments - his … . The sides of the triangle form at the vertices of the triangle ….

c) According to the size of the angle, triangles are ... , ... , ... .

d) Based on the number of equal sides, triangles are ... , ... , ... .

2. Draw

A) right triangle;

b) acute triangle;

c) obtuse triangle;

d) equilateral triangle;

e) scalene triangle;

e) isosceles triangle.

3. Create an assignment on the topic of the lesson for your friends.

More children preschool age know what a triangle looks like. But the kids are already starting to understand what they are like at school. One type is an obtuse triangle. The easiest way to understand what it is is to see a picture of it. And in theory, this is what they call the “simplest polygon” with three sides and vertices, one of which is

Understanding the concepts

In geometry, there are these types of figures with three sides: acute, right and obtuse triangles. Moreover, the properties of these simplest polygons are the same for all. Thus, for all listed species this inequality will be observed. The sum of the lengths of any two sides will necessarily be greater than the length of the third side.

But to be sure that we're talking about It is about the finished figure, and not about a set of individual vertices, that it is necessary to check that the basic condition is met: the sum of the angles of an obtuse triangle is equal to 180 degrees. The same is true for other types of figures with three sides. True, in an obtuse triangle, one of the angles will be even more than 90°, and the remaining two will certainly be acute. In this case, it is the largest angle that will be opposite the longest side. True, these are not all the properties of an obtuse triangle. But even knowing only these features, schoolchildren can solve many problems in geometry.

For every polygon with three vertices, it is also true that by continuing any of the sides, we obtain an angle whose size will be equal to the sum of two non-adjacent internal vertices. The perimeter of an obtuse triangle is calculated in the same way as for other shapes. It is equal to the sum of the lengths of all its sides. To determine this, mathematicians have developed various formulas, depending on what data is initially present.

Correct style

One of the most important conditions for solving geometry problems is the correct drawing. Mathematics teachers often say that it will help not only to visualize what is given and what is required of you, but to get 80% closer to the correct answer. This is why it is important to know how to construct an obtuse triangle. If you just need a hypothetical figure, then you can draw any polygon with three sides so that one of the angles is greater than 90 degrees.

If certain values of the lengths of the sides or degrees of angles are given, then it is necessary to draw an obtuse triangle in accordance with them. In this case, it is necessary to try to depict the angles as accurately as possible, calculating them using a protractor, and display the sides in proportion to the conditions given in the task.

Main lines

Often, it is not enough for schoolchildren to know only what certain figures should look like. They cannot limit themselves to information only about which triangle is obtuse and which is right. The mathematics course requires that their knowledge of the basic features of figures should be more complete.

So, every schoolchild should understand the definition of bisector, median, perpendicular bisector and height. In addition, he must know their basic properties.

Thus, bisectors divide an angle in half, and the opposite side into segments that are proportional to the adjacent sides.

The median divides any triangle into two equal in area. At the point at which they intersect, each of them is divided into 2 segments in a 2: 1 ratio, when viewed from the vertex from which it emerged. In this case, the large median is always drawn to its smallest side.

No less attention is paid to height. This is perpendicular to the side opposite the corner. The height of an obtuse triangle has its own characteristics. If it is drawn from a sharp vertex, then it does not end up on the side of this simplest polygon, but on its continuation.

The perpendicular bisector is the line segment that extends from the center of the triangle's face. Moreover, it is located at a right angle to it.

Working with Circles

At the beginning of studying geometry, it is enough for children to understand how to draw an obtuse triangle, learn to distinguish it from other types and remember its basic properties. But for high school students this knowledge is no longer enough. For example, on the Unified State Exam there are often questions about circumscribed and inscribed circles. The first of them touches all three vertices of the triangle, and the second has one common point with all sides.

Constructing an inscribed or circumscribed obtuse triangle is much more difficult, because to do this you first need to find out where the center of the circle and its radius should be. By the way, necessary tool In this case, not only a pencil with a ruler, but also a compass will become available.

The same difficulties arise when constructing inscribed polygons with three sides. Mathematicians have developed various formulas that allow them to determine their location as accurately as possible.

Inscribed triangles

As stated earlier, if a circle passes through all three vertices, then it is called a circumcircle. Its main property is that it is unique. To find out how the circumscribed circle of an obtuse triangle should be located, you must remember that its center is at the intersection of the three bisectoral perpendiculars that go to the sides of the figure. If in an acute-angled polygon with three vertices this point will be inside it, then in an obtuse-angled polygon it will be outside it.

Knowing, for example, that one of the sides of an obtuse triangle is equal to its radius, you can find the angle that lies opposite the known face. Its sine will be equal to the result of dividing the length of the known side by 2R (where R is the radius of the circle). That is, the sin of the angle will be equal to ½. This means that the angle will be equal to 150°.

If you need to find the circumradius of an obtuse triangle, then you will need information about the length of its sides (c, v, b) and its area S. After all, the radius is calculated like this: (c x v x b) : 4 x S. By the way, it doesn’t matter , what type of figure you have: a scalene obtuse triangle, isosceles, right- or acute-angled. In any situation, thanks to the above formula, you can find out the area of a given polygon with three sides.

Circumscribed triangles

You also often have to work with inscribed circles. According to one formula, the radius of such a figure, multiplied by ½ the perimeter, will be equal to the area of the triangle. True, to figure it out you need to know the sides of an obtuse triangle. After all, in order to determine ½ the perimeter, you need to add their lengths and divide by 2.

To understand where the center of a circle inscribed in an obtuse triangle should be, it is necessary to draw three bisectors. These are the lines that bisect the corners. It is at their intersection that the center of the circle will be located. In this case, it will be equidistant from each side.

The radius of such a circle inscribed in an obtuse triangle is equal to the quotient (p-c) x (p-v) x (p-b): p. In this case, p is the semi-perimeter of the triangle, c, v, b are its sides.

Perhaps the most basic, simple and interesting figure in geometry is the triangle. In the 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 are less than zero, then the triangle is in any case obtuse. 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 ideal view, where all the 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 straight line distance 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 internal corners, 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 basic properties that different types of triangles have. In the future, it is important to understand what you are dealing with when solving a problem.

Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.

Consider the geometric shapes and find the “extra” one among them (Fig. 1).