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Types of triangles, angles and sides. Properties of a triangle

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).

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 side - 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 is a triangle 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.

Bibliography

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. Mathematics lessons: Methodological recommendations for teachers. 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.

Nsportal.ru ().
Prosv.ru ().
Do.gendocs.ru ().

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.

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. In 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.
Each external angle has a value that is obtained by adding two internal angles 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 acute angles must equal 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 different types of triangles.

Consider the 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 side - 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 is a triangle 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.

Bibliography

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. Mathematics lessons: Methodological recommendations for teachers. 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.

Nsportal.ru ().
Prosv.ru ().
Do.gendocs.ru ().

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.

Standard designations

Triangle with vertices A, B And C is designated as (see figure). A triangle has three sides:

The lengths of the sides of a triangle are indicated by lowercase Latin letters (a, b, c):

A triangle has the following angles:

The angle values at the corresponding vertices are traditionally denoted by Greek letters (α, β, γ).

Signs of equality of triangles

A triangle on the Euclidean plane can be uniquely determined (up to congruence) by the following triplets of basic elements:

a, b, γ (equality on two sides and the angle lying between them);
a, β, γ (equality on the side and two adjacent angles);
a, b, c (equality on three sides).

Signs of equality of right triangles:

along the leg and hypotenuse;
on two legs;
along the leg and acute angle;
along the hypotenuse and acute angle.

Some points in the triangle are “paired”. For example, there are two points from which all sides are visible at either an angle of 60° or an angle of 120°. They're called Torricelli dots. There are also two points whose projections onto the sides lie at the vertices of a regular triangle. This - Apollonius points. Points and such are called Brocard points.

Direct

In any triangle, the center of gravity, the orthocenter and the center of the circumcircle lie on the same straight line, called Euler's line.

The straight line passing through the center of the circumcircle and the Lemoine point is called Brocard axis. The Apollonius points lie on it. The Torricelli point and the Lemoine point also lie on the same line. The bases of the external bisectors of the angles of a triangle lie on the same straight line, called axis of external bisectors. The intersection points of the lines containing the sides of an orthotriangle with the lines containing the sides of the triangle also lie on the same line. This line is called orthocentric axis, it is perpendicular to the Euler straight line.

If we take a point on the circumcircle of a triangle, then its projections onto the sides of the triangle will lie on the same straight line, called Simson's straight this point. Simson's lines of diametrically opposite points are perpendicular.

Triangles

A triangle with vertices at the bases drawn through a given point is called cevian triangle this point.
A triangle with vertices in the projections of a given point onto the sides is called sod or pedal triangle this point.
A triangle with vertices at the second points of intersection of lines drawn through the vertices and a given point with the circumscribed circle is called circumferential triangle. The circumferential triangle is similar to the sod triangle.

Circles

Inscribed circle- a circle touching all three sides of the triangle. She's the only one. The center of the inscribed circle is called incenter.
Circumcircle- a circle passing through all three vertices of the triangle. The circumscribed circle is also unique.
Excircle- a circle touching one side of the triangle and the continuation of the other two sides. There are three such circles in a triangle. Their radical center is the center of the inscribed circle of the medial triangle, called Spiker's point.

The midpoints of the three sides of a triangle, the bases of its three altitudes and the midpoints of the three segments connecting its vertices with the orthocenter lie on one circle called circle of nine points or Euler circle. The center of the nine-point circle lies on the Euler line. A circle of nine points touches an inscribed circle and three excircles. The point of tangency between the inscribed circle and the circle of nine points is called Feuerbach point. If from each vertex we lay outwards of the triangle on straight lines containing the sides, orthoses equal in length to the opposite sides, then the resulting six points lie on the same circle - Conway circle. Three circles can be inscribed in any triangle in such a way that each of them touches two sides of the triangle and two other circles. Such circles are called Malfatti circles. The centers of the circumscribed circles of the six triangles into which the triangle is divided by medians lie on one circle, which is called circumference of Lamun.

A triangle has three circles that touch two sides of the triangle and the circumcircle. Such circles are called semi-inscribed or Verrier circles. The segments connecting the points of tangency of the Verrier circles with the circumcircle intersect at one point called Verrier's point. It serves as the center of a homothety, which transforms a circumcircle into an inscribed circle. The points of contact of the Verrier circles with the sides lie on a straight line that passes through the center of the inscribed circle.

The segments connecting the points of tangency of the inscribed circle with the vertices intersect at one point called Gergonne point, and the segments connecting the vertices with the points of tangency of the excircles are in Nagel point.

Ellipses, parabolas and hyperbolas

Inscribed conic (ellipse) and its perspector

An infinite number of conics (ellipses, parabolas or hyperbolas) can be inscribed into a triangle. If we inscribe an arbitrary conic into a triangle and connect the tangent points with opposite vertices, then the resulting straight lines will intersect at one point called prospect bunks. For any point of the plane that does not lie on a side or on its extension, there is an inscribed conic with a perspector at this point.

The described Steiner ellipse and the cevians passing through its foci

You can inscribe an ellipse into a triangle, which touches the sides in the middle. Such an ellipse is called inscribed Steiner ellipse(its perspective will be the centroid of the triangle). The circumscribed ellipse, which touches the lines passing through the vertices parallel to the sides, is called described by the Steiner ellipse. If we transform a triangle into a regular triangle using an affine transformation (“skew”), then its inscribed and circumscribed Steiner ellipse will transform into an inscribed and circumscribed circle. The Chevian lines drawn through the foci of the described Steiner ellipse (Scutin points) are equal (Scutin’s theorem). Of all the described ellipses, the described Steiner ellipse has the smallest area, and of all the inscribed ellipses, the inscribed Steiner ellipse has the largest area.

Brocard ellipse and its perspector - Lemoine point

An ellipse with foci at Brocard points is called Brocard ellipse. Its perspective is the Lemoine point.

Properties of an inscribed parabola

Kiepert parabola

The prospects of the inscribed parabolas lie on the described Steiner ellipse. The focus of an inscribed parabola lies on the circumcircle, and the directrix passes through the orthocenter. A parabola inscribed in a triangle and having Euler's directrix as its directrix is called Kiepert parabola. Its perspector is the fourth point of intersection of the circumscribed circle and the circumscribed Steiner ellipse, called Steiner point.

Kiepert's hyperbole

If the described hyperbola passes through the point of intersection of the heights, then it is equilateral (that is, its asymptotes are perpendicular). The intersection point of the asymptotes of an equilateral hyperbola lies on the circle of nine points.

Transformations

If the lines passing through the vertices and some point not lying on the sides and their extensions are reflected relative to the corresponding bisectors, then their images will also intersect at one point, which is called isogonally conjugate the original one (if the point lay on the circumscribed circle, then the resulting lines will be parallel). Many pairs of remarkable points are isogonally conjugate: the circumcenter and the orthocenter, the centroid and the Lemoine point, the Brocard points. The Apollonius points are isogonally conjugate to the Torricelli points, and the center of the inscribed circle is isogonally conjugate to itself. Under the action of isogonal conjugation, straight lines transform into circumscribed conics, and circumscribed conics into straight lines. Thus, the Kiepert hyperbola and the Brocard axis, the Jenzabek hyperbola and the Euler straight line, the Feuerbach hyperbola and the line of centers of the inscribed and circumscribed circles are isogonally conjugate. The circumcircles of the triangles of isogonally conjugate points coincide. The foci of inscribed ellipses are isogonally conjugate.

If, instead of a symmetrical cevian, we take a cevian whose base is as distant from the middle of the side as the base of the original one, then such cevians will also intersect at one point. The resulting transformation is called isotomic conjugation. It also converts straight lines into described conics. The Gergonne and Nagel points are isotomically conjugate. Under affine transformations, isotomically conjugate points are transformed into isotomically conjugate points. With isotomic conjugation, the described Steiner ellipse will go into the infinitely distant straight line.

If in the segments cut off by the sides of the triangle from the circumcircle, we inscribe circles touching the sides at the bases of the cevians drawn through a certain point, and then connect the tangent points of these circles with the circumcircle with opposite vertices, then such straight lines will intersect at one point. A plane transformation that matches the original point to the resulting one is called isocircular transformation. The composition of isogonal and isotomic conjugates is the composition of an isocircular transformation with itself. This composition is a projective transformation, which leaves the sides of the triangle in place, and transforms the axis of the external bisectors into a straight line at infinity.

If we continue the sides of a Chevian triangle of a certain point and take their points of intersection with the corresponding sides, then the resulting points of intersection will lie on one straight line, called trilinear polar starting point. The orthocentric axis is the trilinear polar of the orthocenter; the trilinear polar of the center of the inscribed circle is the axis of the external bisectors. Trilinear polars of points lying on a circumscribed conic intersect at one point (for a circumscribed circle this is the Lemoine point, for a circumscribed Steiner ellipse it is the centroid). The composition of an isogonal (or isotomic) conjugate and a trilinear polar is a duality transformation (if a point isogonally (isotomically) conjugate to a point lies on the trilinear polar of a point, then the trilinear polar of a point isogonally (isotomically) conjugate to a point lies on the trilinear polar of a point).

Cubes

Ratios in a triangle

Note: in this section, , are the lengths of the three sides of the triangle, and , are the angles lying respectively opposite these three sides (opposite angles).

Triangle inequality

In a non-degenerate triangle, the sum of the lengths of its two sides is greater than the length of the third side, in a degenerate triangle it is equal. In other words, the lengths of the sides of a triangle are related by the following inequalities:

The triangle inequality is one of the axioms of metrics.

Triangle Angle Sum Theorem

Theorem of sines

where R is the radius of the circle circumscribed around the triangle. It follows from the theorem that if a< b < c, то α < β < γ.

Cosine theorem

Tangent theorem

Other ratios

Metric ratios in a triangle are given for:

Solving triangles

Calculating the unknown sides and angles of a triangle based on the known ones has historically been called “solving triangles.” The above general trigonometric theorems are used.

Area of a triangle

Special cases Notation

For the area the following inequalities are valid:

Calculating the area of a triangle in space using vectors

Let the vertices of the triangle be at points , , .

Let's introduce the area vector . The length of this vector is equal to the area of the triangle, and it is directed normal to the plane of the triangle:

Let us set , where , , are the projections of the triangle onto the coordinate planes. Wherein

and similarly

The area of the triangle is .

An alternative is to calculate the lengths of the sides (using the Pythagorean theorem) and then using Heron's formula.

Triangle theorems

Desargues's theorem: if two triangles are perspective (the lines passing through the corresponding vertices of the triangles intersect at one point), then their corresponding sides intersect on the same line.

Sonda's theorem: if two triangles are perspective and orthologous (perpendiculars drawn from the vertices of one triangle to the sides opposite the corresponding vertices of the triangle, and vice versa), then both centers of orthology (the points of intersection of these perpendiculars) and the center of perspective lie on the same straight line, perpendicular to the perspective axis (straight line from Desargues' theorem).