What figures are called equal. Two geometric figures are called equal if they can be combined
Back forward
Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested this work please download the full version.
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. By different 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.
Exercise: 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 equal?
c) Squares with equal sides are equal?
d) Prove that the parallelograms formed by the intersection of two strips of the same width under different angles slopes towards 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 the quadrilateral ABCD, the point E is the midpoint of AB, connected to the vertex D, and F is the midpoint of CD, to the vertex B. Prove that the area of the quadrilateral EBFD is 2 times less area 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 the area of the triangle RMK more area 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 different years, in particular from study guide"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.
Figures are called equal if their shape and size are the same. From this definition it follows, for example, that if the given rectangle and square have equal areas, then they still do not become equal figures, since these are different figures in shape. Or, two circles definitely have the same shape, but if their radii are different, then these are also not equal figures, since their sizes do not match. Equal figures are, for example, two segments of the same length, two circles with the same radius, two rectangles with pairwise equal sides (the short side of one rectangle is equal to the short side of the other, the long side of one rectangle is equal to the long side of the other).
It can be difficult to tell by eye whether shapes that have the same shape are equal. Therefore, to determine the equality of simple figures, they are measured (using a ruler, compass). Segments have length, circles have radius, rectangles have length and width, squares have only one side. It should be noted here that not all figures can be compared. It is impossible, for example, to determine the equality of lines, since any line is infinite and, consequently, all lines can be said to be equal to each other. The same goes for rays. Although they have a beginning, they have no end.
If we are dealing with complex (arbitrary) figures, then it can even be difficult to determine whether they have the same shape. After all, figures can be inverted in space. Look at the picture below. It is difficult to say whether these figures are identical in shape or not.
Thus, it is necessary to have a reliable principle for comparing figures. He is like this: equal figures when superimposed on each other coincide.
To compare the two depicted figures with an overlay, a tracing paper (transparent paper) is applied to one of them and the shape of the figure is copied (copied) onto it. They try to superimpose a copy on the tracing paper on the second figure so that the figures coincide. If this succeeds, then the given figures are equal. If not, then the figures are not equal. When applied, the tracing paper can be rotated as you like, as well as flipped.
If you can cut out the figures themselves (or they are separate flat objects, and not drawn), then tracing paper is not needed.
When studying geometric figures, one can notice many of their features associated with the equality of their parts. So, if you fold a circle along the diameter, then its two halves will be equal (they will overlap). If you cut a rectangle diagonally, you get two right triangle. If one of them is rotated 180 degrees clockwise or counterclockwise, then it will coincide with the second. That is, the diagonal divides the rectangle into two equal parts.
Target: formation of the concept of “equal figures”.
- to form the ability to fix the concept of “equal figures”, to fix 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 on a sea voyage in search of 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)
Exercise: split the figures in the envelope into pairs so that only one sign changes - the size.
Were you able to pair all the items? (No)
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!
what is the angle called? What figures are called equal? Explain how to compare two segments? what point is calledthe 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 calledthe 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-angled, which triangle is called obtuse-angled, which 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?
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...
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?
What figures are called equal?
Shapes are called equal, which match when superimposed.
A common mistake to this question is the answer, which mentions the equal sides and angles of a geometric figure. However, this does not take into account that the sides of a geometric figure are not necessarily straight. Therefore, only the coincidence of geometric figures when superimposed can be a sign of their equality.
In practice, this is easy to check using the overlay, they must match.
Everything is very simple and accessible, usually equal figures can be seen immediately.
Equal are those figures that have the same geometry parameters. These parameters are: the length of the sides, the size of the angles, the thickness.
The easiest way to understand that the figures are equal is by using an overlay. If the sizes of the figures are the same, they are called equal.
Equal name only those geometric shapes that have exactly the same parameters:
1) perimeter;
2) area;
4) dimensions.
That is, if one figure is superimposed on another, then they will coincide.
It is a mistake to assume that if the figures have the same perimeter or area, then they are equal. In fact, geometric figures that have the same area are called equal.
Figures are called equal if they coincide when superimposed on each other. Equal figures have the same size, shape, area and perimeter. But figures that are equal in area may not be equal to each other.
In geometry, according to the rules, equal figures must have the same area and perimeter, that is, they must have absolutely the same shape and size. And they must match exactly when they are superimposed on each other. If there are any discrepancies, then these figures can no longer be called equal.
Figures can be called equal provided that they completely coincide when superimposed on each other, i.e. they have the same size, shape and therefore area and perimeter, as well as other characteristics. Otherwise, it is impossible to talk about the equality of the figures.
The very word equal contains the essence.
These are figures that are completely identical to each other. That is, they completely match. If the figure is put one on one then the figures will overlap themselves from all sides.
They are the same, that is, they are equal.
Unlike equal triangles (to determine which it is enough to fulfill one of the conditions - signs of equality), equal figures are called those that have the same not only shape, but also dimensions.
To determine whether one figure is equal to another, you can use the overlay method. In this case, the figures must match both sides and corners. These will be equal figures.
Only such figures can be equal, which, when superimposed, completely coincide with the sides and corners. In fact, for all the simplest polygons, the equality of their area also indicates the equality of the figures themselves. Example: a square with side a will always be equal to another square with the same side a. The same applies to rectangles and rhombuses - if their sides are equal to the sides of another rectangle, they are equal. A more complex example: triangles will be congruent if they have equal sides and corresponding angles. But these are only special cases. In more general cases, the equality of the figures is nevertheless proved by superposition, and this superposition in planimetry is pompously called movement.
