Rectangle definition and characteristics, brief description. What is a rectangle? Special cases of a rectangle
Rectangle is a quadrilateral in which each angle is right.
Proof
The property is explained by the action of feature 3 of the parallelogram (that is, \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 on 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.
Of all the figures (only of parallelograms!), only the rectangle 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 angles of a parallelogram). And \angle A = \angle C , \angle B = \angle D .
We conclude that \angle A = \angle B = \angle C = \angle D. They are all 90^(\circ) . In total - 360^(\circ) .
Proven!
6. The square of a diagonal is equal to the sum of the squares of its two adjacent sides.
This property is true due to 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 point of intersection of the diagonals divides them in half.
AO = BO = CO = DO
9. The point of intersection of the diagonals is the center of the rectangle and the circumcircle.
10. The sum of all angles is 360 degrees.
\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^(\circ)
11. All angles of a rectangle are right.
\angle ABC = \angle BCD = \angle CDA = \angle DAB = 90^(\circ)
12. The diameter of a circle circumscribed around a rectangle is equal to the diagonal of the rectangle.
13. You can always describe a circle around a rectangle.
This property is true due to the fact that the sum of the opposite angles 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 equal side lengths (it is a square).
Lesson on the topic “Rectangle and its properties”
Lesson objectives:
Repeat the concept of a rectangle, based on the knowledge acquired by students in the mathematics course of grades 1–6.
Consider the properties of a rectangle as a special type of parallelogram.
Consider a particular property of a rectangle.
Show the application of properties to problem solving.
During the classes.
I O organizational moment.
Inform the purpose of the lesson, the topic of the lesson.
II Learning new material.
Repeat:
1. What figure is called a parallelogram?
2. What properties does a parallelogram have?
● Introduce the concept of a rectangle.
Which parallelogram can be called a rectangle?
Definition: A rectangle is a parallelogram in which all angles are right.(slide 3)
This means that since a rectangle is a parallelogram, it has all the properties of a parallelogram. Since the rectangle has a different name, it must have its own property (slide 4).
● Student activity (independent): Explore the sides, angles and diagonals of a parallelogram and a rectangle, recording the results in a table.
Parallelogram
Rectangle
Parties
Angles
Diagonals
Draw a conclusion: The diagonals of the rectangle are equal.
● This output is a particular property of the rectangle:
Theorem. D The diagonals of the rectangle are equal.
Given: ABCD – rectangle,
AC and BD diagonals.
Prove: AC = BD
Proof:
1) Consider ∆ ACD and ∆ ABD:
A)
AD C =
D AB = 90°,
b) A D– general,
c) AB = C D – opposite sides of the rectangle,
Therefore, the triangles are equal on two sides.
2) Since the triangles are equal, then AC = BD.
● Let's consider the properties of a rectangle, knowing that it is a parallelogram.
Property 1: the sum of the angles of a rectangle is 360°.
Proof: a) since a rectangle has four angles of 90°, their sum is 360°.
b) since a rectangle is a quadrilateral, the sum of the angles of a quadrilateral is (n – 2) ∙180° = (4 – 2) ∙180° = 2∙180° = 360°.
Property 2: opposite sides of the rectangle are equal.
Proof: a) since a rectangle is a parallelogram, and a parallelogram has opposite sides equal, then the opposite sides of a rectangle will also be equal.
How else can you prove this fact?
b) if we draw a diagonal AC, then from the equality of right triangles ABC and CDAnd (according to the hypotenuse and the acute angle) the equality of the opposite sides of the rectangle will follow.
Property 3: The diagonals of the rectangle intersect and are bisected by the intersection point.
Proof: a) since a rectangle is a parallelogram, and in a parallelogram the diagonals intersect and are divided in half by the intersection point, then the diagonals of a rectangle intersect and are divided in half by the intersection point.
Is there another proof of this property?
b) Yes, through the equality of triangles AOB and D OS (along a side and two adjacent angles)
Property 4: The bisector of an angle of a rectangle cuts off an isosceles triangle from it.
Proof: a) since a rectangle is a parallelogram, and in a parallelogram the bisector of an acute angle cuts off an isosceles triangle from it, then in a rectangle the bisector of any angle cuts off an isosceles triangle from it.
Is there any other way to prove this property?
b) It is possible. Consider the right triangle ABC and prove the equality of the angles BAK and BKA. Then we can conclude that the sides AB and BC are equal.
All properties are proved using the properties of a parallelogram.
We found that a rectangle has five properties:
III Consolidation of the studied material.
Class assignments: 1. Find the perimeter of a rectangle (orally)
a)b)
Solution:
a) P = (6+4)∙2, P = 20(dm) (opposite sides of the rectangle are equal)
b) because the diagonals of the rectangle are equal, then ∆ M ОK and ∆ M ОN are isosceles, OB and OA are medians, therefore they are also heights. Then 2BO = MN = 8, 2AO = MK = 4.
Р = (8 + 4)∙2, Р = 24(dm)
2. Find the sides of the rectangle, knowing that its perimeter is 24 cm.
Solution: 1) ∆АВМ is isosceles, since AM is a bisector,
means AB = VM.
2) 24 = (AB + VM + MS) ∙2,
12 = AB + VM + MS,
12 = VM + VM +MS,
12 = MS + 2∙VM.
3)
3 MV = 9, MV = 3, MS = 6
4) AB = CD = 3, AD = BC = 3 +6 = 9
Answer: 3 cm, 9 cm, 3 cm, 9 cm.
№ 403 (textbook)
Given: ABCO -rectangle, D = 30°,
means C D = 0.5AC = 6 cm.
2) AB = C D = 6 cm.
3) In a rectangle, the diagonals are equal and are divided in half by the point of intersection, i.e. AO = BO = 6 cm.
4) P(aov) = AO + VO + AB = 6 +6+ 6 = 18cm.
Answer: 18 cm.
IV Summing up the lesson.
A rectangle has the following properties:
1. The sum of the angles of a rectangle is 360°.
2. Opposite sides of the rectangle are equal.
3. The diagonals of the rectangle intersect and are divided in half by the intersection point.
4. The bisector of the angle 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. 399, 401 a), 404
At home, consider the sign of a rectangle yourself.
The rectangle is unique in its simplicity. Based on this figure, students begin to learn the basics of geometry. Therefore, in high school they get lost, not knowing the basic properties and characteristics of a rectangle, in vain considering this figure to be too simple.
Rectangle
The definition of a rectangle has been known since elementary school: it is a parallelogram in which all angles are equal to 90 degrees. The question arises: what is a parallelogram?
Despite the tricky name, this shape is as simple as a rectangle. A parallelogram is a convex quadrilateral whose sides are pairwise equal and parallel.
In the definition, be sure to highlight the word convex. Because convex and non-convex quadrilaterals are clearly separated in geometry. Moreover, non-convex figures are not studied at all in the school mathematics course, since they are much more unpredictable in their properties.
Rice. 1. Convex quadrilaterals
A rectangle is a special case of a parallelogram. Moreover, there are other special cases of a parallelogram, for example, a rhombus; So are other special cases of a rectangle - a square. Therefore, before you can prove that a figure is a rectangle, you need to prove that it is a parallelogram.
Rectangle Properties
The properties of a rectangle can be divided into two groups: the properties of a parallelogram and the properties of a rectangle.
Properties of a parallelogram:
- Opposite sides are equal and parallel in pairs.
- Opposite angles are equal.
Rice. 2. Properties of a parallelogram
Rectangle properties:
- All angles are equal to 90 degrees, which comes from the definition of a figure.
- The diagonal of a rectangle divides the figure into two small equal right triangles. This property is easy to prove. The triangles will be rectangular, as they will include one 90-degree angle. In this case, the diagonal will be a common side, and the legs will be equal, since the opposite sides of the rectangle are pairwise equal and parallel.
- The diagonals of a rectangle are equal.
Rice. 3. Beam
Rectangle signs
A rectangle has only three main features:
- On the corner. If one of the angles of a parallelogram is 90 degrees, then the parallelogram is a rectangle.
- If three angles of a quadrilateral are equal to 90 degrees, then the quadrilateral is a rectangle. Please note that in this case there is no need to prove that we have a parallelogram. It is enough to know the values of the angles of a quadrilateral.
- Diagonally: If the diagonals of a parallelogram are equal, then such a parallelogram is a rectangle.
Pay attention to which figure the feature is applied to, this is important in the proof.
What is the difference between a sign and a property? A sign is a difference by which a figure can be distinguished from others. Like a person's name. You see a friend, remember his name and immediately know what to expect from him. But expectations from a person are already properties. Properties can only be applied after you have proven that this or that figure is in front of you. And for this proof we need signs.
What have we learned?
We learned what a parallelogram is. We talked about special cases of a parallelogram, including the most common one - a rectangle. The properties and characteristics of a rectangle were identified. We noticed that some of the signs are valid for any quadrilateral, and some only for a parallelogram.
Test on the topic
Article rating
Average rating: 4.1. Total ratings received: 268.
Sections: Primary School
Subject: Types of quadrangles. Rectangle
- Ensure that students acquire knowledge about the different types of quadrilaterals and rectangles.
- Develop the ability to classify facts, draw conclusions, build a rectangle and distinguish it from a number of quadrangles.
- Cultivating learning motives and a positive attitude towards classes.
Lesson type – combined.
The type of lesson is a didactic game.
Teaching methods and techniques: dialogical and heuristic methods:
- organization of work in pairs;
- frontal work;
- operational form of knowledge testing (special cards);
- demonstration of visual aids;
- work in teams.
Equipment:
- overhead projector;
- poster with types of quadrangles;
- visual aids for the fairy tale;
- signal cards;
- punched cards for each student with prepared tables;
- rectangle blanks;
- scissors, rulers, pencils, drawing triangles;
- magnetic board;
- rectangles with numbers;
- handouts (red rectangles to encourage respondents);
- record player.
During the classes
I. Updating previous knowledge (5 minutes)Today in class we will take a trip to an amazing country. Geometry:
– Who knows what the word “geometry” means in Greek?
“Geo” – earth, “metry” – measurement.
This science appeared in Greece.
We will be accompanied on our journey (the teacher shows a fairy-tale hero) by an amazing hero - a wizard.
– He encrypted all of you, and you will travel under encrypted numbers.
-Who recognized him? (Old man Hottabych.)
– Who wrote the book “Old Man Hottabych”? (Lagin.)
Old Man Hottabych is a very old wizard and his knowledge is outdated, so he came to your lesson and wants to find out what modern children are studying now. Help the wizard figure it out.
– What is shown on the board? (Geometric figures.)
– Determine which 2 groups you could divide these geometric shapes into? (Triangles and quadrilaterals.)
Fill out card No. 1. Indicate the numbers of triangles and quadrilaterals. All children indicate numbers on the card.
At this time, 2 students record their answers on the board.
– Indicate on the second card the numbers of triangles by angles (obtuse, rectangular, acute) and sides (equilateral and isosceles).
The work is carried out according to the options, and then they exchange cards and carry out mutual checking in pairs.
II. Formation of new concepts and methods of action(20 minutes)
1) Today our hero and I will get acquainted with the types of quadrilaterals, namely; with a rectangle, let's learn how to draw it and distinguish it from other shapes. There are many triangles and quadrilaterals in geometry. Here's what some of them look like:
TYPES OF QUADAGONS
– Which of them do you already know?
Children name the species they know.
– What do these figures have in common that unites them into one group?
(4 sides, 4 corners, 4 vertices.)
– How does one type differ from another? (Lengths of the sides and features of the corners.)
The teacher draws the children's attention to the table and says the definitions.
- Square - a rectangle with all sides equal.
- Trapezoid – a quadrilateral in which only 2 opposite sides are parallel (translation: “table”).
- Parallelogram - a quadrilateral whose opposite sides are parallel and equal. - a parallelogram with all sides equal.
- Irregular quadrilateral - a figure whose sides are not equal and not parallel.
2) Help Hottabych find similar ones from a series of quadrangles (1 3 5).
– What are the names of the angles of figures 1, 3, 5? (Direct.)
– What would you call these figures? (Rectangles.)
– Try to tell me what a rectangle is?
A rectangle is a geometric figure in which all angles are right and opposite sides are equal.
– What are the vertices of the rectangle ABCD? (A, B, C, D are vertices.)
- What about the corners? (<АВД, <ВДС, <ДСА, <САВ)
- Sides? (AV, VD, SD, SA)
– Do you think a rectangle is a necessary geometric figure or not (yes).
A fairy tale will help you see this.
3) Fairy tale “Useful rectangle”.
The rectangle was jealous of the square.
- I'm so clumsy. If I rise to my full height, I will become long and narrow. Like this:
– And if I lie on my side, I’ll be short and fat:
- And you always remain the same - standing, sitting, and lying down.
“Yes,” the square said proudly. For me, all sides are equal, not like some people, sometimes it’s big-headed, sometimes it’s pancake-pancake. And one day this happened:
Old man Hottabych got lost in the forest. He didn’t have a flying carpet, his beard was wet from the rain, and he couldn’t get out of the forest. He walked through the thicket and came across a square and a rectangle.
– Can I climb on top of you and see where my home is? - he asked the square.
Hottabych first climbed to one side of the square, but did not see anything because the tops of the trees were in the way. Then the wizard asked the square to turn over to the other side, but, as you know, all sides of a square are equal, so again he did not see anything.
- Citizen Square, help me at least get across the river. The square approached the river and tried to touch the other bank. BUT...splash!.
– Maybe I can help you? – suggested a modest rectangle.
He stood up to his full height and Hottabych climbed on him and
was higher than the trees. In the distance he saw his house and knew where to go. Then the rectangle lay on its side and became a bridge. Hottabych crossed the river along the rectangle, helped him up and, thanking the rectangle, went home.
And the square, which was drying on the shore after swimming, said
rectangle:
– It turns out that you are a useful figure
- Well, what are you talking about! – the rectangle smiled modestly.
It’s just that my sides are different lengths: 2 are long, 2 are short. Sometimes this can be very convenient.
– What rectangular objects do you see in your classroom?
4) There is a special drawing triangle with which you can determine right angles in a geometric figure. Try to determine experimentally which of these shapes are rectangles.
CARD #3.
– How did the drawing triangle help you in this search?
Children identify and name the numbers of the figures (2,4). They demonstrate on the board how the drawing triangle helped them in their definition.
5) Fizminutka(song “Twice two is four”).
Your teacher will be happy
Look at your
Children stand near their desks
Show it to everyone
Put your hands forward
And then vice versa
The result was an airplane
Let's take flight
Inseparable friends / 2 times
Square, rectangle,
Inseparable friends
Geometry and schoolboy
6) Draw a rectangle using segments and a drawing triangle:
Children draw in their notebooks, and then with an explanation at the board.
Draw a 4 cm segment. Combine the side of the triangle with the segment and build a right angle, set aside the segment, etc.
III. Formation of skills (18 minutes)1. Draw a rectangle, knowing that one side is 2 cm and the other is 4 cm larger.
Task analysis:
– Can you immediately draw a rectangle? (No)
- Why? (We don’t know the length of the second side.)
- How to find the length of the second side? (2+4=6).
A team of 4 people is working.
2. You have rectangle blanks with sides of 8 cm and 4 cm. They need to be cut into 4 identical triangles, and then made into a square. How to do it?
3. Old Man Hottabych wants to make sure that you were attentive and learned what we talked about. On his behalf, I ask questions, and you show the answer using signal cards: Yes – green, No – red.
1) Is it true that if a figure has 4 corners, 4 sides, 4 vertices, then it can be called a quadrilateral? (Yes)
2) Is a rectangle a type of quadrilateral? (Yes)
3) Is it true that the opposite sides of a rectangle are not equal? (No)
4) Is it correct that a square can be called a rectangle and a quadrilateral? (Yes)
4. Graphic dictation
Mark point A, from it downwards at a right angle draw a segment 2 cm long and mark its end with point B. From B to the right at a right angle draw a segment 4 cm long and mark the end with point C. Draw a segment 2 cm long upward at a right angle and mark point D. Complete the figure yourself, which we paid a lot of attention to in the lesson.
-What figure is this? (rectangle)
5. Find 3 quadrilaterals in the drawing:
6. Riddles.
Having solved the riddles, you will find out what our guest wants to tell you.
– What figure are we talking about?
He's been my friend for a long time,
Every angle in it is right.
All four sides
Same length.
I'm glad to introduce him to you.
- What is his name? ( Square)
– What kind of figure can say that about himself?
You're on me, you're on him,
Look at all of us.
We have everything, we have everything
On three sides and three corners,
And as many peaks
And three times - difficult things,
We will do it three times. ( Triangle)
– What types of quadrilaterals do you know?
– What shape is called a rectangle?
V. Homework.Come up with a fairy tale or crossword puzzle about geometric shapes.
Bibliography:
- V. Volina “Feast of the Number”, Moscow, Bustard 1997
- A.M. Pyshkalo “Methodology for teaching the elements of geometry in primary school”, Education, 1980.
- Magazine “Zavuch”, No. 1, 2000, Fomin A.A. “Compliance with pedagogical requirements as a factor that increases the professional competence of a modern teacher,” p. 21.
- Magazine “Primary School”, No. 2, 2001 “Geometry”, p.15.
- Newspaper “Primary School”, No. 3, 1997 “Geometry”, p. 4.
Lesson Objectives
To consolidate students' knowledge on the topic rectangle;
Continue introducing students to the definitions and properties of a rectangle;
Teach schoolchildren to use the acquired knowledge on this topic when solving problems;
Develop interest in the subject of mathematics, attention, logical thinking;
Develop the ability to self-analysis and discipline.
Lesson Objectives
To repeat and consolidate students’ knowledge about such a concept as a rectangle, building on the knowledge acquired in previous grades;
Continue to improve schoolchildren’s knowledge about the properties and characteristics of rectangles;
Continue to develop skills in the process of solving tasks;
Arouse interest in mathematics lessons;
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 studied topics about rectangles. Now let's refresh our memory and remember what kind of figure it is that is called a rectangle.
A rectangle is a parallelogram whose four angles are right and equal to 90 degrees.
A rectangle is 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 according to Euclidean geometry, then for a quadrilateral to be considered a rectangle, it is necessary that in this geometric figure at least three angles are right. It follows from this 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.
If a regular rectangle has all sides 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, in addition to equal sides, has all right angles.
To prove the involvement of any geometric figure in a rectangle, it is sufficient 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. the diagonals of the geometric figure must have the same length;
3. all angles of a geometric figure must be equal to 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 subtypes, with their own special properties and characteristics.
Exercise: Name the geometric shapes that belong to rectangles.
Rectangle and its properties
Now let's remember 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;
A rectangle has equal opposite sides and angles;
A circle can be circumscribed around any rectangle, and 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 equal rectangles.
2. Draw a circle around the rectangle, the diameter of which will be equal to the diagonal of the rectangle.
3. Is it possible to inscribe a circle in a rectangle so that it touches all its sides, but provided that this rectangle is not a square?
Rectangle signs
The parallelogram will be a rectangle provided:
1. if at least one of its angles is right;
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 the non-opposite sides.
It's interesting to know
Did you know that if you draw bisectors of the corners 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.
Did 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, the canvas of which depicted a black rectangle with the peculiar name “Battle of the Negroes in the Tunnel”.
This idea with a black rectangle inspired other cultural figures. The French writer and humorist Alphonse Allais released a whole series of his works and over time a rectangular landscape in a radical red color appeared called “Harvesting tomatoes on the shores of the Red Sea by apoplectic cardinals,” which also did not have any image.
Exercise
1. Name a property that is inherent only 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 this is so, then prove why?
4. List the quadrilaterals that are rectangles.
5. State the properties of a rectangle.
Historical fact
Euclid's rectangle
Did you know that the Euclid rectangle, which is called the golden ratio, for a long period of time was for any building of religious significance, a perfect and proportional basis for construction in those days. With its help, most of the Renaissance buildings and classical temples in Ancient Greece were built.
A “golden” rectangle is usually called a geometric rectangle, the ratio of the larger side to the smaller side is equal to the golden ratio.
This ratio of the sides of this rectangle was 382 to 618, or approximately 19 to 31. The Euclidian rectangle, at that time, was the most expedient, convenient, safe and regular rectangle of all geometric shapes. Due to this characteristic, the Euclidian rectangle, or approximations to it, was 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 the usual, standard shape of the house, symbolizing the woman who owns this house.
Subjects > Mathematics > Mathematics 8th grade