Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.
Examine the geometric shapes and find the “extra” among them (Fig. 1).
Rice. 1. Illustration for example
We see that figures No. 1, 2, 3, 5 are quadrangles. Each of them has its own name (Fig. 2).
Rice. 2. Quadrangles
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 straight line, and three segments connecting these points in pairs.
The points are called triangle vertices, 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. Triangles are classified according to the angle 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 right-angled 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, i.e. greater than 90° (Fig. 6).
Rice. 6. Obtuse Triangle
According to the number of equal sides, triangles are equilateral, isosceles, scalene.
An isosceles triangle is a triangle in which two sides are equal (Fig. 7).
Rice. 7. Isosceles triangle
These sides are called lateral, the third side - basis. In an isosceles triangle, the angles at the base are equal.
Isosceles triangles are acute and obtuse(Fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is called, 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 triangle is called versatile, in which all three sides have different lengths (Fig. 10).
Rice. 10. Scalene triangle
Complete the task. Divide 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: #2, #6.
Obtuse triangles: #4, #5.
These triangles are divided 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.
Review the drawings.
Think about what piece of wire each triangle is made of (fig. 12).
Rice. 12. Illustration for the task
You can argue like this.
The first piece of wire is divided into three equal parts, so you can make equilateral triangle. It is shown third in the figure.
The second piece of wire is divided into three different parts, so you can make a scalene triangle out of it. It is shown first in the picture.
The third piece of wire is divided into three parts, where the two parts are the same length, so you can make an isosceles triangle out of it. It is shown second in the figure.
Today in the lesson we got acquainted with different types of triangles.
Bibliography
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
- M.I. Moreau. Math Lessons: Guidelines for the teacher. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Mathematics: Verification work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Finish the phrases.
a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.
b) The points are called … , segments - his … . The sides of a triangle form at the vertices of a triangle ….
c) According to the size of the angle, triangles are ..., ..., ....
d) According to the number of equal sides, triangles are ..., ..., ....
2. Draw
a) a right triangle
b) an acute triangle;
c) an obtuse triangle;
d) an equilateral triangle;
e) scalene triangle;
e) an isosceles triangle.
3. Make a task on the topic of the lesson for your comrades.
Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.
Examine the geometric shapes and find the “extra” among them (Fig. 1).
Rice. 1. Illustration for example
We see that figures No. 1, 2, 3, 5 are quadrangles. Each of them has its own name (Fig. 2).
Rice. 2. Quadrangles
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 straight line, and three segments connecting these points in pairs.
The points are called triangle vertices, 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. Triangles are classified according to the angle 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 right-angled 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, i.e. greater than 90° (Fig. 6).
Rice. 6. Obtuse Triangle
According to the number of equal sides, triangles are equilateral, isosceles, scalene.
An isosceles triangle is a triangle in which two sides are equal (Fig. 7).
Rice. 7. Isosceles triangle
These sides are called lateral, the third side - basis. In an isosceles triangle, the angles at the base are equal.
Isosceles triangles are acute and obtuse(Fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is called, 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 triangle is called versatile, in which all three sides have different lengths (Fig. 10).
Rice. 10. Scalene triangle
Complete the task. Divide 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: #2, #6.
Obtuse triangles: #4, #5.
These triangles are divided 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.
Review the drawings.
Think about what piece of wire each triangle is made of (fig. 12).
Rice. 12. Illustration for the task
You can argue like this.
The first piece of wire is divided into three equal parts, so you can make an equilateral triangle out of it. It is shown third in the figure.
The second piece of wire is divided into three different parts, so you can make a scalene triangle out of it. It is shown first in the picture.
The third piece of wire is divided into three parts, where the two parts are the same length, so you can make an isosceles triangle out of it. It is shown second in the figure.
Today in the lesson we got acquainted with different types of triangles.
Bibliography
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
- M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Nsportal.ru ().
- Prosv.ru ().
- Do.gendocs.ru ().
Homework
1. Finish the phrases.
a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.
b) The points are called … , segments - his … . The sides of a triangle form at the vertices of a triangle ….
c) According to the size of the angle, triangles are ..., ..., ....
d) According to the number of equal sides, triangles are ..., ..., ....
2. Draw
a) a right triangle
b) an acute triangle;
c) an obtuse triangle;
d) an equilateral triangle;
e) scalene triangle;
e) an isosceles triangle.
3. Make a task on the topic of the lesson for your comrades.
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.
At right triangles there are 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.
When deciding geometric problems it is useful to follow such an algorithm. While reading the task statement, it is necessary
- Make a drawing. The drawing should correspond as much as possible to the condition of the problem, so its main task is to help find the solution
- Apply all the data from the task condition to the drawing
- write out everything geometric concepts, which are encountered in the problem
- Recall all the theorems that relate to this concept
- Put on the drawing all the relationships between the elements geometric figure, which follow from these theorems
For example, if the task contains the words bisector of the angle of a triangle, you need to remember the definition and properties of the bisector and designate equal or proportional segments and angles in the drawing.
In this article, you will find the basic properties of a triangle that you need to know in order to successful solution tasks.
TRIANGLE.
Area of a triangle.
1. ,
here - an arbitrary side of the triangle, - the height lowered to this side.
2.
,
here and are arbitrary sides of the triangle, is the angle between these sides:
3. Heron formula:
Here - the lengths of the sides of the triangle, - the semiperimeter of the triangle,
4. ,
here - the semiperimeter of the triangle, - the radius of the inscribed circle.
Let be the lengths of the tangent segments.
Then Heron's formula can be written in the following form:
5.
6. ,
here - the lengths of the sides of the triangle, - the radius of the circumscribed circle.
If a point is taken on a side of a triangle that divides this side in the ratio m:n, then the segment connecting this point with the vertex of the opposite angle divides the triangle into two triangles, the areas of which are related as m:n:
The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.
Triangle median
This is a line segment that connects the vertex of the triangle with the midpoint of the opposite side.
Triangle medians intersect at one point and share the intersection point in a ratio of 2:1, counting from the top.
The intersection point of the medians of a regular triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger one is equal to the radius of the circumscribed circle.
The radius of the circumscribed circle is twice the radius of the inscribed circle: R=2r
Median length arbitrary triangle
,
here - the median drawn to the side - the lengths of the sides of the triangle.
Bisector of a triangle
This is a segment of the bisector of any angle of a triangle, connecting the vertex of this angle with the opposite side.
Bisector of a triangle divides the side into segments proportional to the adjacent sides:
Triangle bisectors intersect at one point, which is the center of the inscribed circle.
All points on the bisector of an angle are equidistant from the sides of the angle.
Triangle Height
This is a segment of the perpendicular, lowered from the vertex of the triangle to the opposite side, or its continuation. In an obtuse triangle, the altitude drawn from the vertex of an acute angle lies outside the triangle.
The heights of a triangle intersect at one point, which is called the triangle's orthocenter.
To find the height of a triangle drawn to the side, you need to find its area in any way possible, and then use the formula:
Center of a circle circumscribed about a triangle, lies at the point of intersection of the perpendicular bisectors drawn to the sides of the triangle.
The radius of the circumscribed circle of a triangle can be found using the following formulas:
Here, are the lengths of the sides of the triangle, and is the area of the triangle.
,
where is the length of the side of the triangle, is the opposite angle. (This formula follows from the sine theorem).
triangle inequality
Each side of the triangle is less than the sum and greater than the difference of the other two.
The sum of the lengths of any two sides is always greater than the length of the third side:
Opposite the larger side lies a larger angle; opposite the larger angle lies the larger side:
If , then vice versa.
Sine theorem:
The sides of a triangle are proportional to the sines of the opposite angles:
Cosine theorem:
square side of a triangle is equal to the sum squares of the other two sides without doubling the product of these sides by the cosine of the angle between them:
Right triangle
- It is a triangle with one of the angles equal to 90°.
Sum sharp corners of a right triangle is 90°.
The hypotenuse is the side that lies opposite the 90° angle. The hypotenuse is the longest side.
Pythagorean theorem:
the square of the hypotenuse is equal to the sum of the squares of the legs: 
The radius of a circle inscribed in a right triangle is
,
here - the radius of the inscribed circle, - the legs, - the hypotenuse:
Center of a circle circumscribed about a right triangle lies in the middle of the hypotenuse:
Median of a right triangle drawn to the hypotenuse equal to half of the hypotenuse.
Definition of sine, cosine, tangent and cotangent of a right triangle see
The ratio of elements in a right triangle:
The square of the altitude of a right triangle drawn from a vertex right angle, is equal to the product of the projections of the legs on the hypotenuse:
The square of the leg is equal to the product of the hypotenuse and the projection of the leg to the hypotenuse:
Leg lying against the corner equal to half the hypotenuse:
Isosceles triangle.
Bisector isosceles triangle drawn to the base is the median and the height.
In an isosceles triangle, the angles at the base are equal.
Top angle.
I - sides
And - angles at the base.
Height, bisector and median.
Attention! The height, bisector and median drawn to the lateral side do not match.
right triangle
(or equilateral triangle ) is a triangle, all sides and angles of which are equal to each other.
Area of an equilateral triangle is equal to
where is the length of the side of the triangle.
Center of a circle inscribed in an equilateral triangle, coincides with the center of the circle circumscribed about an equilateral triangle and lies at the point of intersection of the medians.
Intersection point of medians of an equilateral triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger one is equal to the radius of the circumscribed circle.
If one of the angles of an isosceles triangle is 60°, then the triangle is regular.
Middle line of the triangle
This is a segment that connects the midpoints of two sides.
In the figure, DE is the midline of triangle ABC.
The midline of the triangle is parallel to the third side and equal to half of it: DE||AC, AC=2DE
External corner of a triangle
This is the angle adjacent to any angle of the triangle.
An exterior angle of a triangle is equal to the sum of two angles not adjacent to it.
Trigonometric functions of an external angle:
Signs of equality of triangles:
1 . If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.
2 . If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.
3 If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.
Important: since in a right triangle two angles are obviously equal, then for equality of two right triangles only two elements are required to be equal: two sides, or a side and an acute angle.
Signs of similarity of triangles:
1 . If two sides of one triangle are proportional to two sides of another triangle, and the angles included between these sides are equal, then these triangles are similar.
2 . If three sides of one triangle are proportional to three sides of another triangle, then these triangles are similar.
3 . If two angles of one triangle are equal to two angles of another triangle, then these triangles are similar.
Important: In similar triangles, similar sides lie opposite equal angles.
Theorem of Menelaus
Let the line intersect the triangle , where is the point of its intersection with the side , is the point of its intersection with the side , and is the point of its intersection with the extension of the side . Then
