A rectangle, its properties and area. Rectangle. Complete lessons – Knowledge Hypermarket
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Parallelogram, rectangle, rhombus, square (2019)
1. Parallelogram
Compound word "parallelogram"? And behind it lies a very simple figure.
Well, that is, we took two parallel lines:
Crossed by two more:
And inside there is a parallelogram!
What properties does a parallelogram have?
Properties of a parallelogram.
That is, what can you use if the problem is given a parallelogram?
The following theorem answers this question:
Let's draw everything in detail.
What does it mean first point of the theorem? And the fact is that if you HAVE a parallelogram, then you will certainly
The second point means that if there IS a parallelogram, then, again, certainly:
Well, and finally, the third point means that if you HAVE a parallelogram, then be sure to:
Do you see what a wealth of choice there is? What to use in the problem? Try to focus on the question of the task, or just try everything one by one - some “key” will do.
Now let’s ask ourselves another question: how can we recognize a parallelogram “by sight”? What must happen to a quadrilateral so that we have the right to give it the “title” of a parallelogram?
Several signs of a parallelogram answer this question.
Signs of a parallelogram.
Attention! Begin.
Parallelogram.
Please note: if you found at least one sign in your problem, then you definitely have a parallelogram, and you can use all the properties of a parallelogram.
2. Rectangle
I think that it will not be news to you at all that
First question: is a rectangle a parallelogram?
Of course it is! After all, he has - remember, our sign 3?
And from here, of course, it follows that in a rectangle, like in any parallelogram, the diagonals are divided in half by the point of intersection.
But the rectangle also has one distinctive property.
Rectangle property
Why is this property distinctive? Because no other parallelogram has equal diagonals. Let's formulate it more clearly.
Please note: in order to become a rectangle, a quadrilateral must first become a parallelogram, and then demonstrate the equality of the diagonals.
3. Diamond
And again the question: is a rhombus a parallelogram or not?
With every right - a parallelogram, because it has and (remember our feature 2).
And again, since a rhombus is a parallelogram, then it must have all the properties of a parallelogram. This means that in a rhombus, opposite angles are equal, opposite sides are parallel, and the diagonals bisect at the point of intersection.
Properties of a rhombus
Look at the picture:
As in the case of a rectangle, these properties are distinctive, that is, for each of these properties we can conclude that this is not just a parallelogram, but a rhombus.
Signs of a diamond
And again, pay attention: there must be not just a quadrilateral whose diagonals are perpendicular, but a parallelogram. Make sure:
No, of course, although its diagonals are perpendicular, and the diagonal is the bisector of the angles and. But... diagonals are not divided in half by the point of intersection, therefore - NOT a parallelogram, and therefore NOT a rhombus.
That is, a square is a rectangle and a rhombus at the same time. Let's see what happens.
Is it clear why? - rhombus is the bisector of angle A, which is equal to. This means it divides (and also) into two angles along.
Well, it's quite clear: the diagonals of a rectangle are equal; The diagonals of a rhombus are perpendicular, and in general, a parallelogram of diagonals is divided in half by the point of intersection.
AVERAGE LEVEL
Properties of quadrilaterals. Parallelogram
Properties of a parallelogram
Attention! Words " properties of a parallelogram"mean that if in your task There is parallelogram, then all of the following can be used.
Theorem on the properties of a parallelogram.
In any parallelogram:
Let's understand why this is all true, in other words WE'LL PROVE theorem.
So why is 1) true?
If it is a parallelogram, then:
- lying like criss-cross
- lying like crosses.
This means (according to criterion II: and - general.)
Well, that’s it, that’s it! - proved.
But by the way! We also proved 2)!
Why? But (look at the picture), that is, precisely because.
Only 3 left).
To do this, you still have to draw a second diagonal.
And now we see that - according to the II characteristic (angles and the side “between” them).
Properties proven! Let's move on to the signs.
Signs of a parallelogram
Recall that the parallelogram sign answers the question “how do you know?” that a figure is a parallelogram.
In icons it's like this:
Why? It would be nice to understand why - that's enough. But look:
Well, we figured out why sign 1 is true.
Well, it's even easier! Let's draw a diagonal again.
Which means:
AND It's also easy. But...different!
Means, . Wow! But also - internal one-sided with a secant!
Therefore the fact that means that.
And if you look from the other side, then - internal one-sided with a secant! And therefore.
Do you see how great it is?!
And again simple:
Exactly the same, and.
Pay attention: if you found at least one sign of a parallelogram in your problem, then you have exactly parallelogram and you can use everyone properties of a parallelogram.
For complete clarity, look at the diagram:
Properties of quadrilaterals. Rectangle.
Rectangle properties:
Point 1) is quite obvious - after all, sign 3 () is simply fulfilled
And point 2) - very important. So, let's prove that
This means on two sides (and - general).
Well, since the triangles are equal, then their hypotenuses are also equal.
Proved that!
And imagine, equality of diagonals is a distinctive property of a rectangle among all parallelograms. That is, this statement is true^
Let's understand why?
This means (meaning the angles of a parallelogram). But let us remember once again that it is a parallelogram, and therefore.
Means, . Well, of course, it follows that each of them! After all, they have to give in total!
So they proved that if parallelogram suddenly (!) the diagonals turn out to be equal, then this exactly a rectangle.
But! Pay attention! This is about parallelograms! Not just anyone a quadrilateral with equal diagonals is a rectangle, and only parallelogram!
Properties of quadrilaterals. Rhombus
And again the question: is a rhombus a parallelogram or not?
With full right - a parallelogram, because it has (Remember our feature 2).
And again, since a rhombus is a parallelogram, it must have all the properties of a parallelogram. This means that in a rhombus, opposite angles are equal, opposite sides are parallel, and the diagonals bisect at the point of intersection.
But there are also special properties. Let's formulate it.
Properties of a rhombus
Why? Well, since a rhombus is a parallelogram, then its diagonals are divided in half.
Why? Yes, that's why!
In other words, the diagonals turned out to be bisectors of the corners of the rhombus.
As in the case of a rectangle, these properties are distinctive, each of them is also a sign of a rhombus.
Signs of a diamond.
Why is this? And look,
That means both These triangles are isosceles.
To be a rhombus, a quadrilateral must first “become” a parallelogram, and then exhibit feature 1 or feature 2.
Properties of quadrilaterals. Square
That is, a square is a rectangle and a rhombus at the same time. Let's see what happens.
Is it clear why? A square - a rhombus - is the bisector of an angle that is equal to. This means it divides (and also) into two angles along.
Well, it's quite clear: the diagonals of a rectangle are equal; The diagonals of a rhombus are perpendicular, and in general, a parallelogram of diagonals is divided in half by the point of intersection.
Why? Well, let's just apply the Pythagorean theorem to...
SUMMARY AND BASIC FORMULAS
Properties of a parallelogram:
- Opposite sides are equal: , .
- Opposite angles are equal: , .
- The angles on one side add up to: , .
- The diagonals are divided in half by the point of intersection: .
Rectangle properties:
- The diagonals of the rectangle are equal: .
- A rectangle is a parallelogram (for a rectangle all the properties of a parallelogram are fulfilled).
Properties of a rhombus:
- The diagonals of a rhombus are perpendicular: .
- The diagonals of a rhombus are the bisectors of its angles: ; ; ; .
- A rhombus is a parallelogram (for a rhombus all the properties of a parallelogram are fulfilled).
Properties of a square:
A square is a rhombus and a rectangle at the same time, therefore, for a square all the properties of a rectangle and a rhombus are fulfilled. And.
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 for 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 Oorganizational 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 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 |
|
Diagonals |
Draw a conclusion: The diagonals of the rectangle are equal.
● This output is a private property of the rectangle:
Theorem. D The diagonals of the 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 the rectangle, knowing that its perimeter is 24 cm.
1)ACD - rectangular, CAD = 30°,
means CD = 0.5AC = 6 cm.
2) AB = CD = 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. 000, 401 a), 404 (slide 16)
At home, consider the sign of a rectangle yourself.
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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 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 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 gradeA rectangle is Firstly geometric flat figure. It consists of four points that are connected to each other by two pairs of equal segments that intersect perpendicularly only at these points.
A rectangle is defined through a parallelogram. In other words, a rectangle is a parallelogram whose angles are all right angles, that is, equal to 90 degrees. In Euclidean geometry, if a geometric figure has 3 out of 4 angles equal to 90 degrees, then the fourth angle is automatically equal to 90 degrees and such a figure can be called a rectangle. From the definition of a parallelogram it is clear that a rectangle is many varieties of this figure on a plane. It follows that the properties of a parallelogram also apply to a rectangle. For example: in a rectangle, opposite sides are equal in length. When constructing a diagonal in a rectangle, it will split the figure into two identical triangles. This is the basis of the Pythagorean theorem, which states that the square of the hypotenuse in a right triangle is equal to the sum of the squares of its legs. If all sides of a regular rectangle are equal, then such a rectangle is called a square. A square is also defined as a rhombus in which all its sides are equal and all its angles are right angles.
