Lesson Objectives
To consolidate students' knowledge on the topic of the rectangle;
Continue to introduce students to the definitions and properties of a rectangle;
To teach schoolchildren to use the acquired knowledge on this topic while solving problems;
To develop interest in the subject of mathematics, attention, logical thinking;
Cultivate the ability to introspection and discipline.
Lesson objectives
To repeat and consolidate the knowledge of schoolchildren about such a concept as a rectangle, starting from the knowledge gained in previous classes;
Continue to improve the knowledge of schoolchildren about the properties and features of rectangles;
Continue to develop skills in the process of solving tasks;
Generate interest in mathematics lessons;
To cultivate interest in the exact sciences and a positive attitude towards mathematics lessons.
Lesson Plan
1. Theoretical part, general information, definitions.
2. Repetition of the theme "Rectangles".
3. Properties of a rectangle.
4. Signs of a rectangle.
5. Interesting Facts from the life of triangles.
6. Golden rectangle, general concepts.
7. Questions and tasks.
What is a rectangle
In previous classes, you have already learned topics about rectangles. Now let's refresh our memory and remember what kind of figure it is, which is called a rectangle.
A rectangle is a parallelogram whose four angles are right and equal to 90 degrees.
A rectangle is such a geometric figure, consisting of 4 sides and four right angles.
Opposite sides of a rectangle are always equal.
If we consider the definition of a rectangle in Euclidean geometry, then for a quadrilateral to be considered a rectangle, it is necessary that in this geometric figure at least three angles be right. From this it follows that the fourth angle will also be ninety degrees.
Although it is clear that when the sum of the angles of a quadrilateral does not have 360 degrees, then this figure is not a rectangle.
In the case when all sides of a regular rectangle are equal to each other, then such a rectangle is called a square.
In some cases, a square can act as a rhombus if such a rhombus, except for equal sides, has all right angles.
To prove the involvement of any geometric figure in a rectangle, it is enough that this geometric figure meets at least one of these requirements:
1. the square of the diagonal of this figure must be equal to the sum of the squares of 2 sides that have a common point;
2. diagonals of a geometric figure must have the same length;
3. all angles of a geometric figure must be ninety degrees.
If these conditions meet at least one requirement, then you have a rectangle.
A rectangle in geometry is the main basic figure, which has many subspecies, with its own special properties and characteristics.
Exercise: Name the geometric shapes that are related to rectangles.
Rectangle and its properties
Now let's recall the properties of a rectangle:
A rectangle has all its diagonals equal;
A rectangle is a parallelogram with parallel opposite sides;
The sides of the rectangle will also be its heights;
The rectangle has equal opposite sides and corners;
A circle can be circumscribed around any rectangle, moreover, the diagonal of the rectangle will be equal to the diameter of the circumscribed circle.
The diagonals of a rectangle divide it into 2 equal triangles;
Following the Pythagorean theorem, the square of the diagonal of a rectangle is equal to the sum of the squares of its 2 non-opposite sides;
Exercise:
1. A rectangle has two possibilities in which it can be divided into 2 equal rectangles. Draw two rectangles in your notebook and divide them so that you get 2 rectangles equal to each other.
2. Describe a circle around the rectangle, the diameter of which will be equal to the diagonal of the rectangle.
3. Can a circle be inscribed in a rectangle so that it touches all its sides, but on the condition that this rectangle is not a square?
Rectangle Features
A parallelogram will be a rectangle if:
1. if it has at least one of the right angles;
2. if all four of its angles are right;
3. if opposite sides are equal;
4. if at least three angles are right;
5. if its diagonals are equal;
6. if the square of the diagonal is equal to the sum of the squares of non-opposite sides.
It's interesting to know
Did you know that if you draw angle bisectors in a rectangle that has uneven adjacent sides, then when they intersect, you will end up with a rectangle.
But if the drawn bisector of a rectangle intersects one of its sides, then it cuts off an isosceles triangle from this rectangle.
Do you know that even before Malevich painted his outstanding “Black Square”, in 1882, at an exhibition in Paris, a painting by Paul Bilo was presented, on the canvas of which a black rectangle was depicted with the peculiar name “Battle of the Negroes in the Tunnel”.
Such an idea with a black rectangle inspired other cultural figures. The French humorist Alphonse Allais published a whole series of his works and over time a rectangular landscape appeared in radical red called "Harvesting tomatoes on the Red Sea coast by apoplectic cardinals", which also had no image.
Exercise
1. Name a property that is unique to a rectangle?
2. What is the difference between an arbitrary parallelogram and a rectangle?
3. Is it true that any rectangle can be a parallelogram? If so, please prove why?
4. List the quadrilaterals that are rectangles.
5. Formulate the properties of the rectangle.
historical fact
Euclid's rectangle
Do you know that Euclid's rectangle, which is called the golden ratio, for a long period of time was for any building of religious significance, the perfect and proportional basis of construction in those days. With his help, most of the buildings of the Renaissance and classical temples in Ancient Greece were built.
A "golden" rectangle is usually called such a geometric rectangle, the ratio of the larger side of which to the smaller one is equal to the golden ratio.
This ratio of the sides of this rectangle was 382 to 618, or approximately 19 to 31. Euclid's rectangle, at that time, was the most expedient, convenient, safe and regular rectangle of all geometric shapes. Due to this characteristic, Euclid's rectangle, or an approximation to it, has been used throughout. It was used in houses, paintings, furniture, windows, doors and even books.
Among the Navajo Indians, the rectangle was compared with the female form, since it was considered normal, standard form house, symbolizing the woman who owns this house.
Subjects > Mathematics > Mathematics Grade 8Rectangle is a quadrilateral in which every corner is a right angle.
Proof
The property is explained by the action of feature 3 of the parallelogram (i.e. \angle A = \angle C , \angle B = \angle D )
2. Opposite sides are equal.
AB = CD,\enspace BC = AD
3. Opposite sides are parallel.
AB \parallel CD,\enspace BC \parallel AD
4. Adjacent sides are perpendicular to each other.
AB \perp BC,\enspace BC \perp CD,\enspace CD \perp AD,\enspace AD \perp AB
5. The diagonals of the rectangle are equal.
AC=BD
Proof
According to property 1 the rectangle is a parallelogram, which means AB = CD.
Therefore, \triangle ABD = \triangle DCA along two legs (AB = CD and AD - joint).
If both figures - ABC and DCA are identical, then their hypotenuses BD and AC are also identical.
So AC = BD .
Only a rectangle of all figures (only from parallelograms!) Has equal diagonals.
Let's prove this too.
ABCD is a parallelogram \Rightarrow AB = CD , AC = BD by condition. \Rightarrow \triangle ABD = \triangle DCA already on three sides.
It turns out that \angle A = \angle D (like the corners of a parallelogram). And \angle A = \angle C , \angle B = \angle D .
We deduce that \angle A = \angle B = \angle C = \angle D. They are all 90^(\circ) . The total is 360^(\circ) .
Proven!
6. The square of the diagonal is equal to the sum of the squares of its two adjacent sides.
This property is valid by virtue of the Pythagorean theorem.
AC^2=AD^2+CD^2
7. The diagonal divides the rectangle into two identical right triangles.
\triangle ABC = \triangle ACD, \enspace \triangle ABD = \triangle BCD
8. The intersection point of the diagonals bisects them.
AO=BO=CO=DO
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9. The point of intersection of the diagonals is the center of the rectangle and the circumscribed circle.
10. The sum of all angles is 360 degrees.
\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^(\circ)
11. All corners of the rectangle are right.
\angle ABC = \angle BCD = \angle CDA = \angle DAB = 90^(\circ)
12. The diameter of the circumscribed circle around the rectangle is equal to the diagonal of the rectangle.
13. A circle can always be described around a rectangle.
This property is valid due to the fact that the sum of the opposite corners of a rectangle is 180^(\circ)
\angle ABC = \angle CDA = 180^(\circ),\enspace \angle BCD = \angle DAB = 180^(\circ)
14. A rectangle can contain an inscribed circle and only one if it has the same side lengths (it is a square).
Geography, biology, chemistry, algebra, geometry... Schoolchildren have to deal with a lot of information from a wide variety of sciences. However, there are areas of knowledge in which it is quite easy to understand, having familiarized yourself with their basic laws. Geometry is one of them. To know all the subtleties of this science, you must definitely get acquainted with its basics, axioms. After all, without the foundations in geometry, nowhere.
Definition of a rectangle
A rectangle is a geometric figure with four right angles. The definition is quite simple, but you should not think that the student will not have problems studying such a topic, because there are a number of features here. The dimensions of a rectangle depend on the length of its sides, which are most often denoted by the Latin letters a and b.
Rectangle properties
- the sides lying opposite each other are equal and parallel;
- the diagonals of the figure are equal;
- the intersection point of the diagonals bisects them;
- a rectangle can be divided into two equal
Rectangle Features
There are only three features that a rectangle has. Here they are:
- a parallelogram with equal diagonals is a rectangle;
- a parallelogram with one right angle is a rectangle;
- a quadrilateral with three right angles is a rectangle.
A little more interesting
So, what a rectangle is is now clear, but what role it plays in geometric problems and in practical measurements has yet to be figured out. So, first of all, it must be said that this is the most convenient geometric figure, with which you can divide the area into sections both in open areas and indoors.
What is a rectangle? As you know, it is a quadrilateral. There are many varieties of the latter, among which one can name a trapezoid (only two sides are equal), a parallelogram (opposite sides are parallel), a square (all angles and sides are the same), a rhombus (a parallelogram with equal sides) and others. A special case of a rectangle is a square, in which all angles are right, and the sides are equal.
It is impossible to talk about what a rectangle is without mentioning how to determine its dimensions. This area is considered to be the product of its width and length, and the perimeter, like that of any figure, is equal to the sum of the lengths of all sides. In this case, it is also equal to twice the sum of the length and width, since the opposite sides of the rectangle are equal. Now you know what a rectangle is and what to do with it, solving problems and comprehending the secrets of such a mysterious and mysterious science as geometry.
Lesson on the topic "Rectangle and its properties"
Lesson Objectives:
Repeat the concept of a rectangle, based on the knowledge gained by students in the course of mathematics grades 1 - 6.
Consider the properties of a rectangle as a particular type of parallelogram.
Consider a particular property of a rectangle.
Show the application of properties to problem solving.
During the classes.
I Oorganizing moment.
Inform the purpose of the lesson, the topic of the lesson. (slide 1)
IILearning new material.
· Repeat:
1. What figure is called a parallelogram?
2. What properties does a parallelogram have? (slide 2)
● Introduce the concept of a rectangle.
Which parallelogram can be called a rectangle?
Definition: A rectangle is a parallelogram with all right angles.(slide 3)
So, since a rectangle is a parallelogram, then it has all the properties of a parallelogram. Since the rectangle has a different name, it must have its own property (slide 4).
● Student task (self-guided): Explore the sides, angles, and diagonals of a parallelogram and a rectangle, recording the results in a table.
Parallelogram | Rectangle |
|
Diagonals |
Make a conclusion: the diagonals of the rectangle are equal.
● This output is a private property of the rectangle:
Theorem. D diagonals of a rectangle are equal.(slides 5)
Proof:
1) Consider ∆ACD and ∆ABD:
a) ADC = https://pandia.ru/text/78/059/images/image005_65.jpg" width="120" height="184 src="> 
a) b) 
181">
2. Find the sides of a rectangle knowing that its perimeter is 24 cm.
1) ACD - rectangular, in it CAD \u003d 30 °,
so CD = 0.5AC = 6 cm.
2) AB = CD = 6 cm.
3) In a rectangle, the diagonals are equal and the intersection point is divided in half, i.e. AO \u003d VO \u003d 6 cm.
4) p (aow) \u003d AO + BO + AB \u003d 6 + 6 + 6 \u003d 18 cm.
Answer: 18 cm.
IV Summing up the lesson.
The rectangle has the following properties:
1. The sum of the angles of a rectangle is 360°.
2. Opposite sides of a rectangle are equal.
3. The diagonals of the rectangle intersect and the intersection point is divided in half.
4. The angle bisector of a rectangle cuts off an isosceles triangle from it.
5. The diagonals of the rectangle are equal.
V Homework.
P. 45, questions 12,13. No. 000, 401 a), 404 (slide 16)
At home, consider the sign of a rectangle on your own.
Rectangle … Spelling Dictionary
Parallelogram, quadrilateral, square Dictionary of Russian synonyms. rectangle n., number of synonyms: 4 square (9) ... Synonym dictionary
A term used in the technical analysis of financial market conditions to refer to price movements that fit into a rectangle on a chart. Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B. Modern economic dictionary. 2nd ed., corrected ... Economic dictionary
Glossary of business terms
RECTANGLE, parallelogram, all angles of which are right ... Modern Encyclopedia
A quadrilateral with all right angles... Big Encyclopedic Dictionary
RECTANGLE, four-sided geometric figure (quadrilateral), internal corners which are straight and opposite sides are pairwise parallel and equal. This is a special case of PARALLELOGRAM... Scientific and technical encyclopedic dictionary
RECTANGLE, rectangle, male. (geom.). A quadrilateral in which all angles are right. Dictionary Ushakov. D.N. Ushakov. 1935 1940 ... Explanatory Dictionary of Ushakov
RECTANGLE, a, husband. 1. A quadrilateral with all right angles. 2. The name of the officer's insignia of this form on the buttonholes in the Red Army (from 1924 to 1943). Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
A type of price movement chart in the form of a triangle, used in the technical analysis of financial markets. Dictionary of business terms. Akademik.ru. 2001 ... Glossary of business terms
Books
- Rectangle (+ stickers), Valeria Vilyunova. This sticker book is designed for the youngest readers. At 2 years old, the child is happy to perform exciting tasks by sticking stickers in the right place. This activity is not only…
- Rectangle, Vilyunova V.A. The book "Rectangle" is intended for the smallest readers. With its help, your baby will get acquainted with geometric shapes- a rectangle and a trapezoid, learn to distinguish and name ...
