What figures are called different. Equivalent figures

Target: formation of the concept of “equal figures”.

form the ability to fix the concept “ equal figures”, to fixing the ability to find equal figures;
develop mathematical speech, geometric thinking; train mental operations;
improve counting skills within 9;
educate students in discipline, the ability to work together.

During the classes

1. Organizational moment

Introduction by the teacher.

Pirates are sea robbers, their main goal has always been the search for treasure. We will be good pirates and go to cruise looking for our treasure. I got my hands on an old pirate map.

It is very confusing, many islands are marked on it to confuse the seekers, but you need to get to the island where the treasures are hidden. To find it, we will need to overcome many obstacles. You are ready? Then go.

We will travel by ship.

Let's go to the first island.

2. Oral account

So, following our map, we ended up on an island called “Mental Account”. And to move on, we need to complete the tasks:

Name the neighbors of numbers: 3, 6, 8;

Fill in the blanks:

7,….,….,….,…, 12

10,…,…., 7,….,…,….,…., 2

Solve the example using a number line.

3. Updating knowledge

The next island that we met on the way is “Geometric Island”. He is fraught with his secrets and mysteries that we need to uncover!

Guys need to remember and draw all known to us geometric figures. (Circle, square, rhombus, oval, rectangle)

Look at the picture, what figures are shown?

On what grounds can all figures be divided into groups? (Color, shape, size). Name these groups.

4. Introduction to new material

We successfully coped with the task and can go to the next island. On the third island, I found secret messages for you and me. Everyone has an envelope on their desk. Let's open them and see what kind of test awaits us this time. (Each envelope contains a large and small green square, a large and small blue triangle, a large and small yellow rectangle, two red circles of the same size)

Guys, remember on what grounds all the figures are divided? (Color, shape, size)

The task: split the figures in the envelope into pairs so that only one sign changes - the size.

Were you able to pair all the items? (Not)

Why? (Because the two circles are the same size, color and shape)

Prove that these figures are the same. (Overlay)

Let's think about how such figures can be called? ( From the proposed options, the teacher chooses the concept of “equal figures”)

So, guys, the topic of our lesson is “Equal Figures”. ( Topic is posted on the board

Let's get to know them better. To do this, we need to go to the next island, which is called “Equal Figures”.

Arriving on the island, I immediately noticed various figures on the sand, sketched them, since the wave could wash them away at any moment.

Look at the board, these figures:

If they are equal? ( Children first determine visually equal figures, then the student is called to the board)

How do we know if these figures are really equal or not? (By superimposing one figure on another). A practical action is being taken.

So, what figures can we call equal? (Equal figures are those that match when superimposed).

Let us determine what features of equal figures should coincide.

Under the topic of the lesson, a brief record of the children's reasoning is recorded on the board.

(Equal figures are always the same shape and the same size, and the color may vary)

Do you think figures 1 and 2 are equal?

How do we check it? (Students combine the figures and make sure they are equal)

Do you think figures 2 and 3 are equal? (Similar work in progress)

Guys, are figures 1 and 3 equal?

Why? (They are both equal to figure 2, which means they are equal to each other)

Let's check it with an overlay.

The guys make a conclusion, the teacher briefly fixes on the board 1=2 and 2=3, then 1=3 (If the first figure is equal to the second, and the second to the third, then the first figure is equal to the third)

I have a problem, and if I can't overlay the shapes, for example, they are drawn in a notebook, how can I check if they are equal or not? (You can count by cells)

Let's go to the next island.

5. Primary fastening

Work with the textbook.

1) Page 36 #1. Find equal shapes and color them with the same color . The work is carried out according to the options:

Option 1 - No. 1 a)

Option 2 - No. 1 b)

Guys, you coped with this task, but we cannot continue our journey, the ship stumbled upon a reef, we need to collect it again. Because according to the map, the last island is exactly the one we need!

2) Page 36 #2.

6. Review

You were brave today and were not afraid of the difficult trials that we met on the islands. And as a reward for this, you can become captain-teachers of the ship. But being a captain is not easy, you need to know and be able to do a lot, so try to cope with the following tasks:

1) Students are invited to become a teacher: come up with a task for the drawing, control the implementation, evaluate.

2) Cards are distributed. All errors must be found. Pair check.

8=8 4+3=8 8-2>8-3

7>4 3+1<6 5+1<5+4

3<1 5<5+4 9-7=9-6

7. Lesson summary, reflection

We arrived at the last island, and here is the treasure! Our path was not in vain, because we were rewarded with such treasures!

Guys, how do you understand the phrase “Knowledge is our wealth”?

There are two emoticons on the table in front of you - sad and cheerful. If you are in a good mood, stick a yellow cheerful smiley to the ship, if you are in a bad mood - red.

Now we are experienced travelers and treasure hunters, and next time we will have new adventures! Thanks for the lesson!

In everyday life, we are surrounded by many different objects. Some of them have the same size and the same shape. For example, two identical sheets or two identical bars of soap, two identical coins, etc.

In geometry, figures that have the same size and shape are called equal figures. The figure below shows two figures A1 and A2. To establish the equality of these figures, we need to copy one of them onto a tracing paper. And then move the tracing paper and combine a copy of one shape with another shape. If they are combined, then this means that these figures are the same figures. When this is written A1 \u003d A2 using the usual equal sign.

Determining the equality of two geometric shapes

We can imagine that the first figure was superimposed on the second figure, and not its copy on the tracing paper. Therefore, in the future we will talk about imposing the figure itself, and not its copy, on another figure. Based on the foregoing, we can formulate the definition equality of two geometric figures.

Two geometric figures are called equal if they can be combined by superimposing one figure on another. In geometry, for some geometric shapes (for example, triangles), special signs are formulated, upon fulfillment of which we can say that the figures are equal.

what is the angle called? What figures are called equal? Explain how to compare two segments? what point is called

the middle of the segment?

Which ray is called the angle bisector?

what is the degree measure of an angle?

What figure is called a triangle? What triangles are called equal? Which segment is called the median of a triangle? Which segment is called

the bisector of a triangle? Which segment is called the height of a triangle? Which triangle is called isosceles? Which triangle is called equilateral? Definition of radius, diameter, chord. Give a definition of parallel lines. What angle is called the external angle of a triangle? Which triangle is called acute, which triangle is called obtuse, which is right-angled. What are the sides of a right triangle called? Property of two lines parallel to a third. Theorem on a line intersecting one of the parallel lines. Property of two lines perpendicular to a third

What shape is called a broken line? What are vertex links and polyline length?

Explain what a broken line is called a polygon. What are the vertices, sides, perimeter and diagonals of a polygon? What is a convex polygon?
Explain what angles are called convex angles of a polygon. Derive a formula for calculating the sum of the angles of a convex n-gon. Prove that the sum of the exterior angles of a convex polygon. TAKEN one at each vertex, equals 360 degrees.
What is the sum of the angles of a convex quadrilateral?

1) What shape is called a quadrilateral?

2) What are vertices, angles, sides, diagonals, perimeter of a quadrilateral?
3) What side angles of a quadrilateral are called convex?
4) what is the sum of the angles of a convex quadrilateral?
5) what quadrilateral is called convex?
6) what quadrilateral is called a parallelogram?
7) what properties does a parallelogram have?
8) name the signs of a parallelogram.
9) formulate the properties of a rectangle.
10) what quadrilateral is called a square?
11) formulate the properties of a rhombus.
12) what quadrilateral is called a rhombus?
13) what quadrilateral is called a rectangle?
14) what properties does a square have? please answer briefly...

Geometry Atanasyan 7,8,9 class “Questions answers to questions for repetition to chapter 2 to the textbook of geometry 7-9 class atanasyan Explain what figure

called a triangle.
2. What is the perimeter of a triangle?
3. What triangles are called equal?
4. What is a theorem and proof of a theorem?
5. Explain which segment is called a perpendicular drawn from a given point to a given line.
6. Which segment is called the median of the triangle? How many medians does a triangle have?
7. Which segment is called the bisector of a triangle? How many bisectors does a triangle have?
8. What segment is called the height of the triangle? How many heights does a triangle have?
9. What triangle is called isosceles?
10. What are the names of the sides of an isosceles triangle?
11. What triangle is called an equilateral triangle?
12. Formulate the property of angles at the base of an isosceles triangle.
13. Formulate a theorem on the bisector of an isosceles triangle.
14. Formulate the first sign of equality of triangles.
15. Formulate the second sign of equality of triangles.
16. Formulate the third criterion for the equality of triangles.
17. Define a circle.
18. What is the center of a circle?
19. What is called the radius of a circle?
20. What is called the diameter of a circle?
21. What is called the chord of a circle?

One of the basic concepts in geometry is a figure. This term means a set of points on a plane, limited by a finite number of lines. Some figures can be considered equal, which is closely related to the concept of movement. Geometric figures can be considered not in isolation, but in one or another relationship with each other - their mutual arrangement, contact and fit, position "between", "inside", the ratio expressed in the concepts of "more", "less", "equal" .Geometry studies the invariant properties of figures, i.e. those that remain unchanged under certain geometric transformations. Such a transformation of space, in which the distance between the points that make up a particular figure remains unchanged, is called movement. Movement can act in different ways: parallel translation, identical transformation, rotation around an axis, symmetry relative to a straight line or plane, central, rotational, translational symmetry .

Movement and equal figures

If such a movement is possible that will lead to the combination of one figure with another, such figures are called equal (congruent). Two figures equal to a third are also equal to each other - such a statement was formulated by Euclid, the founder of geometry. The concept of congruent figures can be explained in a simpler language: equal are such figures that completely coincide when superimposed on each other. This is quite easy to determine if the figures are given in the form of certain objects that can be manipulated - for example, they are cut out of paper, therefore at school in the classroom they often resort to this method of explaining this concept. But two figures drawn on a plane cannot be physically superimposed on each other. In this case, the proof of the equality of the figures is the proof of the equality of all the elements that make up these figures: the length of the segments, the size of the angles, the diameter and radius, if we are talking about a circle.

Equivalent and equidistant figures

With equal figures, one should not confuse equal-sized and equally-composed figures - with all the closeness of these concepts.
Equal-sized figures are those that have an equal area if they are figures on a plane, or an equal volume if we are talking about three-dimensional bodies. The coincidence of all the elements that make up these figures is not mandatory. Equal figures will always be equal in size, but not all figures of equal size can be called equal. The concept of equal composition is most often applied to polygons. It implies that polygons can be divided into the same number of respectively equal shapes. Equivalent polygons are always equal area.

Back forward

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Lesson Objectives: Repeat the topic "Area of a parallelogram". Derive the formula for the area of a triangle, introduce the concept of equal-sized figures. Solving problems on the topic "Areas of equal-sized figures."

During the classes

I. Repetition.

1) Orally according to the finished drawing Derive the formula for the area of a parallelogram.

2) What is the relationship between the sides of the parallelogram and the heights dropped on them?

(according to the finished drawing)

the relationship is inversely proportional.

3) Find the second height (according to the finished drawing)

4) Find the area of the parallelogram according to the finished drawing.

Solution:

5) Compare the areas of parallelograms S1, S2, S3. (They have equal areas, all have base a and height h).

Definition: Figures having equal areas are called equal.

II. Problem solving.

1) Prove that any line passing through the point of intersection of the diagonals divides it into 2 equal parts.

Solution:

2) In parallelogram ABCD CF and CE heights. Prove that AD ∙ CF = AB ∙ CE.

3) Given a trapezoid with bases a and 4a. Is it possible to draw straight lines through one of its vertices, dividing the trapezoid into 5 triangles of equal area?

Solution: Can. All triangles are equal.

4) Prove that if we take point A on the side of the parallelogram and connect it to the vertices, then the area of the resulting triangle ABC is equal to half the area of the parallelogram.

Solution:

5) The cake has the shape of a parallelogram. Kid and Carlson divide it like this: Kid points to a point on the surface of the cake, and Carlson cuts the cake into 2 pieces along a straight line passing through this point and takes one of the pieces for himself. Everyone wants a bigger piece. Where should the Kid put an end to?

Solution: At the point of intersection of the diagonals.

6) On the diagonal of the rectangle, a point was chosen and straight lines were drawn through it, parallel to the sides of the rectangle. On opposite sides formed 2 rectangles. Compare their areas.

Solution:

III. Studying the topic "Area of a triangle"

start with a task:

"Find the area of a triangle whose base is a and the height is h."

The guys, using the concept of equal-sized figures, prove the theorem.

Let's build a triangle to a parallelogram.

The area of a triangle is half the area of a parallelogram.

The task: Draw equal triangles.

A model is used (3 colored triangles are cut out of paper and glued at the bases).

Exercise number 474. "Compare the areas of the two triangles into which the given triangle is divided by its median."

Triangles have the same bases a and the same height h. Triangles have the same area

Conclusion: Figures having equal areas are called equal.

Questions for the class:

Are equal figures the same size?
Formulate the opposite statement. Is it true?
Is it true:
a) Are equilateral triangles equal in area?
b) Equilateral triangles with equal sides are equal?
c) Squares with equal sides are equal?
d) Prove that the parallelograms formed by the intersection of two strips of the same width at different angles of inclination to each other are equal. Find the parallelogram of the smallest area formed by the intersection of two strips of the same width. (Show on model: equal width stripes)

IV. Step forward!

Written on the board optional tasks:

1. "Cut the triangle with two straight lines so that you can fold the pieces into a rectangle."

Solution:

2. "Cut the rectangle in a straight line into 2 parts, from which you can make a right triangle."

Solution:

3) A diagonal is drawn in a rectangle. In one of the resulting triangles, a median is drawn. Find the ratios between the areas of figures .

Solution:

Answer:

3. From the Olympiad tasks:

“In quadrilateral ABCD, point E is the midpoint of AB, connected to vertex D, and F is the midpoint of CD, to vertex B. Prove that the area of quadrilateral EBFD is 2 times less than the area of quadrilateral ABCD.

Solution: draw a diagonal BD.

Exercise number 475.

“Draw triangle ABC. Through vertex B, draw 2 straight lines so that they divide this triangle into 3 triangles with equal areas.

Use the Thales theorem (divide AC into 3 equal parts).

V. Task of the day.

For her, I took the extreme right part of the board, on which I write the task of today. The kids may or may not decide. We will not solve this problem in class today. It's just that those who are interested in them can write it off, solve it at home or during a break. Usually, already at recess, many guys begin to solve the problem, if they decide, they show the solution, and I fix it in a special table. In the next lesson, we will definitely return to this problem, devoting a small part of the lesson to solving it (and a new problem can be written on the board).

“A parallelogram is cut into a parallelogram. Divide the rest into 2 equal-sized figures.

Solution: The secant AB passes through the intersection point of the diagonals of the parallelograms O and O1.

Additional problems (from Olympiad problems):

1) “In trapezoid ABCD (AD || BC), vertices A and B are connected to point M, the midpoint of side CD. The area of triangle ABM is m. Find the area of the trapezoid ABCD.

Solution:

Triangles ABM and AMK are equal figures, because AM is the median.
S ∆ABK = 2m, ∆BCM = ∆MDK, S ABCD = S ∆ABK = 2m.

Answer: SABCD = 2m.

2) "In the trapezoid ABCD (AD || BC), the diagonals intersect at the point O. Prove that triangles AOB and COD are equal areas."

Solution:

S ∆BCD = S ∆ABC , because they have a common base BC and the same height.

3) Side AB of an arbitrary triangle ABC is extended beyond vertex B so that BP = AB, side AC is extended beyond vertex A so that AM = CA, side BC is extended beyond vertex C so that KS = BC. How many times is the area of triangle RMK greater than the area of triangle ABC?

Solution:

In a triangle MVS: MA = AC, so the area of triangle BAM is equal to the area of triangle ABC. In a triangle workstation: BP = AB, so the area of the triangle BAM is equal to the area of the triangle ABP. In a triangle ARS: AB = BP, so the area of triangle BAC is equal to the area of triangle BPC. In a triangle VRK: BC \u003d SC, therefore, the area of \u200b\u200bthe triangle VRS is equal to the area of the triangle RKS. In a triangle AVK: BC = SC, so the area of triangle BAC is equal to the area of triangle ASC. In the triangle MSC: MA = AC, so the area of the triangle KAM is equal to the area of the triangle ASC. We get 7 equal triangles. Means,

Answer: The area of triangle MRK is 7 times the area of triangle ABC.

4) Linked parallelograms.

2 parallelograms are located as shown in the figure: they have a common vertex and one more vertex for each of the parallelograms lies on the sides of the other parallelogram. Prove that the areas of parallelograms are equal.

Solution:

And , means,

List of used literature:

Textbook "Geometry 7-9" (authors L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev (Moscow, "Enlightenment", 2003).
Olympiad problems of different years, in particular from the textbook "The best problems of mathematical Olympiads" (compiled by A.A. Korznyakov, Perm, "Knizhny Mir", 1996).
A selection of tasks accumulated over many years of work.