What properties at the rectangle. What is a rectangle? Private rectangle cases. Opposite sides are equal
Objectives lesson
Consolidate the knowledge of students on a rectangle;
Continue to acquaint students with the definitions and properties of the rectangle;
Teach schoolchildren to use the knowledge gained on this topic while solving problems;
Develop interest in mathematics, attention, logical thinking;
Rail the skill to self-analysis and discipline.
Tasks lesson
Repeat and consolidate the knowledge of schoolchildren about such a thing as a rectangle, pushing away from the knowledge gained in previous classes;
Continue to improve the knowledge of schoolchildren about the properties and signs of rectangles;
Continue to form skills in the process of solving tasks;
Cause interest in mathematics lessons;
Rail interest in accurate sciences and a positive attitude towards mathematics lessons.
Lesson plan
1. Theoretical part, general information, definitions.
2. Repeat the topic "Rectangles".
3. Properties of the rectangle.
4. Signs of the 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 the themes about rectangles. Now let's refresh memory and remember what kind of figure, which is called a rectangle.
The rectangle is a parallelogram, four corners of which are straight and equal to 90 degrees.
The rectangle is such a geometric figure consisting of 4-sides and four straight corners.
The opposite sides of the rectangle are always equal.
If we consider the definition of a rectangle in the Euclidean geometry, so that the quadrangle is considered to be a rectangle, it is necessary that in this geometric figure, at least three angles were straight. From this it follows that the fourth corner will also be ninety degrees.
Although it is clear that when the sum of the angles of the quadrangle does not have 360 \u200b\u200bdegrees, then this figure is not a rectangle.
In the case when the right rectangle is equal to each other, then such a rectangle is called the square.
In some cases, the square can act as a rhombus if such a rhombus is other than equal among themselves and all the corners are direct.
To prove the involvement of any geometric figure to a rectangle, sufficiently that this geometric shape corresponds to at least one of these requirements:
1. The square of the diagonal of this figure should be equal to the sum of the squares of the 2-sides, which have a common point;
2. The diagonal of the geometric shape must have the same length;
3. All angles of the geometric shape should be equal to ninety degrees.
If these conditions meet at least one requirement, then you are a rectangle.
The rectangle in geometry is the main basic figure, which has many subspecies, with its special properties and characteristics.
The task: Name the geometric shapes that relate to rectangles.
Rectangle and its properties
And now let's remember the properties of the rectangle:
At the rectangle, all of its diagonals are equal;
The rectangle is a parallelogram with parallel opposite sides;
The sides of the rectangle at the same time will be its heights;
The rectangle has equal opposite sides and angles;
Around every rectangle, you can describe the circle, while the diagonal of the rectangle will be equal to the diameter of the described circle.
The diagonal of the rectangle is separated by 2 equal triangles;
Following the Pythagora theorem, the square of the rectangle diagonal is equal to the sum of the squares of the 2nd not opposite sides;
The task:
1. The rectangle has such two possibilities in which it can be divided into 2 equal rectangles. Distribute two rectangles in the notebook and divide them so that 2 equal rectangles come out.
2. Describe a circle around the rectangle, the diameter of which will be equal to the diagonal of the rectangle.
3. Is it possible to enter a circle into a rectangle so that it concerns all of its sides, but provided that this rectangle is not a square?
Signs of rectangle
The parallelogram will be a rectangle provided:
1. If he has at least one of the corners direct;
2. If all four of its corners are direct;
3. If the opposite sides are equal;
4. If, at least three corners are direct;
5. If his diagonal is equal;
6. If the square is diagonally equal to the sum of the squares of non-opposite sides.
It is interesting to know
Did you know that if in a rectangle, which has uneven adjacent sides, carry out bisector of the corners, then when they are intersection, the rectangle will eventually work.
But if the bisector of the rectangle crosses one of his sides, it cuts off from this rectangle, an equal triangle.
And whether you know that even before Malevich wrote his outstanding "black square", in 1882, at the exhibition in Paris, they presented a picture of Paul Beaulo, on the canvas of which a black rectangle was depicted with a peculiar name "Battle of Blacks in Tunnel".
Such an idea with a black rectangle inspired and other cultural figures. French writer Humorist Alfonso Alla released a whole series of his work and over time there appeared a rectangular landscape in radical red called "Cleaning the yield of tomatoes on the Red Sea shore by apoplexic cardinals", which also had no image.
The task
1. Name the property that is inherent only by the rectangle?
2. What is the difference between an arbitrary parallelogram from a rectangle?
3. Is the approval true that any rectangle modet is a parallelogram? If so, then prove why?
4. List the quadrangles that are rectangles.
5. Formulate the properties of the rectangle.
Historical fact
Rectangle Euclida
Do you know that the rectangle of Euclide, which is called a golden cross section, a long period of time was for any building with religious importance, perfect and proportional to the construction basis in those times. With it, most of the buildings of the Renaissance and classic temples in ancient Greece were built.
The "golden" rectangle is customary to call such a geometric rectangle, the ratio of the main side of which is less equal to the gold proportion.
This ratio of the parties of this rectangle was 382 to 618, or approximately 19 to 31. The rectangle of Euclide, at that time was the most appropriate, convenient, safe and right rectangle from all geometric shapes. Thanks to this characteristic, the rectangle of euclide or approaching it was used everywhere. It was used in homes, paintings, objects of furniture, windows, doors and even books.
Among the Navajo tribe Indians, the rectangle was compared with the female form, as she was considered the usual, standard form of the house, symbolizing a woman who owns this house.
Subject\u003e Mathematics\u003e Mathematics Grade 8Rectangle - parallelogram, in which all corners are direct (equal to 90 degrees). The area of \u200b\u200bthe rectangle is equal to the product of its adjacent sides. The diagonal of the rectangle is equal. The second formula for finding the area of \u200b\u200bthe rectangle comes from the formula of the quadrilateral area through the diagonal.
Rectangle - This is a quadrangle, whose angle is direct.
Square is a special case of a rectangle.
The rectangle has two pairs of equal sides. The length of the longest pairs of the parties is called length rectangle, and the length of the most short - rectangle width.
Properties of rectangles
1. Rectangle is a parallelogram
The property is explained by the action of a sign 3 parallelogram (that is, \\ (\\ angle a \u003d \\ angle c \\), \\ (\\ angle b \u003d \\ angle d \\))
2. The opposite parties are equal
\\ (AB \u003d CD, \\ Enspace BC \u003d AD \\)
3. Opposite sides are parallel
\\ (AB \\ Parallel CD, \\ Enspace BC \\ Parallel AD \\)
4. The adjacent parties are perpendicular to each other
\\ (AB \\ PERP BC, \\ ENSPACE BC \\ PERP CD, \\ Enspace CD \\ PERP AD, \\ ENSPACE AD \\ PERP AB \\)
5. Rectangle diagonals are equal
\\ (AC \u003d BD \\)
According to property 1. The rectangle is a parallelogram, which means \\ (ab \u003d Cd \\).
Hence, \\ (\\ triangle abd \u003d \\ triangle dca \\) For two categories (\\ (ab \u003d CD \\) and \\ (AD \\) - joint).
If both figures - \\ (ABC \\) and \\ (DCA \\) are identical, then their hypotenuses \\ (BD \\) and \\ (AC \\) are also identical.
So, \\ (AC \u003d BD \\).
Only at the rectangle from all figures (only from the parallelograms!) Are equal to the diagonal.
We prove it.
\\ (\\ Rightarrow AB \u003d CD \\), \\ (AC \u003d BD \\) by condition. \\ (\\ Rightarrow \\ triangle abd \u003d \\ triangle dca \\) Already in three parties.
It turns out that \\ (\\ angle a \u003d \\ angle d \\) (as the corners of the parallelogram). And \\ (\\ angle a \u003d \\ angle c \\), \\ (\\ angle b \u003d \\ angle d \\).
We deposit that \\ (\\ angle a \u003d \\ angle b \u003d \\ angle c \u003d \\ angle d \\). All of them \\ (90 ^ (\\ CIRC) \\). In sum - \\ (360 ^ (\\ Circ) \\).
7. Diagonal divides a rectangle on two identical rectangular triangles
\\ (\\ TRIANGLE ABC \u003d \\ TRIANGLE ACD, \\ ENSPACE \\ TRIANGLE ABD \u003d \\ TRIANGLE BCD \\)
8. The intersection point of diagonals shares them in half
\\ (AO \u003d BO \u003d CO \u003d DO \\)
9. The intersection point of diagonals is the center of the rectangle and the circle described
Rectangle is first of allgeometric flat figure. It consists of four points, which are interconnected by two pairs of equal segments, perpendicularly intersecting only at these points.
The rectangle is determined through parallelograms. Differently, the rectangle is a parallelogram whose angles are all straight, that is, equal to 90 degrees. In the geometry of Euclide, if a geometric figure of 3 out of 4 corners are 90 degrees, the fourth corner is automatically equal to 90 degrees and such a figure can be called a rectangle. From the definition of the parallelogram it is clear that the rectangle is a lot of varieties of this figure on the plane. From this it follows that the properties of the parallelogram are applicable to the rectangle. For example: in a rectangle, opposite parties are equal at their length. When building a diagonal in a rectangle, it breaks the figure to two identical triangles. On this and founded the Pythagoreo Theorem, which states that the square of hypotenuses in a rectangular triangle is equal to the sum of the squares of its cathets. If all sides of the right rectangle are equal, then such a rectangle is called a square. The square is also defined as a rhombus, in which all of its parties are equal to each other, and all corners are direct.
In the school program, the geometry lessons have to deal with a variety of quadriclers: rhombuses, parallelograms, rectangles, trapezes, squares. The most first figures for study are rectangle and square.
So what is a rectangle? The definition for grade 2 of the secondary school will look like this: this is a quadricle, who has all four corners direct. It is easy to imagine how the rectangle looks like: this is a figure with 4 straight corners and parties, pairwise parallel to each other.
In contact with
How to understand by solving the next geometric task, with what kind of quadricle are we dealing with? There are three main featuresFor which you can unmistakably determine that we are talking about a rectangle. Let's call them:
- the figure is a quadrilateral, three angle of which are 90 °;
- the presented quadricle is a parallelogram with equal diagonals;
- pollogram, which has at least one straight corner.
It is interesting to know: what is a convex, its features and signs.
Since the rectangle is a parallelogram (i.e., a quadricon with pairwise parallel opposite sides), then all its properties and signs will be performed for it.
Formulas for calculating the length of the parties
In a rectangle The opposite parties are equal and mutually parallel. A longer side is customized with a length (denoted a), shorter - width (designated b). In the rectangle, the lengths are the lengths of AB and CD, and the width - AC and B. D. They are also perpendicular to the grounds (i.e. are altitudes).
To find the sides, you can use the formulas listed below. They adopted conventions: A - the length of the rectangle, B is its width, D is a diagonal (segment connecting the vertices of two angles lying opposite each other), S is the area of \u200b\u200bthe figure, P is the perimeter, α is the angle between the diagonal and length, β - acute angle, which is formed by both diagonals. Ways to find the lengths of the sides:
- Using the diagonal and known side: a \u003d √ (D ² - B ²), B \u003d √ (D ² - a ²).
- According to the area of \u200b\u200bthe figure and one of its sides: a \u003d s / b, b \u003d s / a.
- Using the perimeter and the known side: a \u003d (p - 2 b) / 2, B \u003d (P - 2 A) / 2.
- Through the diagonal and the angle between it and the length: a \u003d d sinα, b \u003d d cosα.
- Through the diagonal and the angle β: a \u003d d sin 0.5 β, B \u003d D cos 0.5 β.
Perimeter and Square
The perimeter of the quadricle is called The sum of all his parties. To calculate the perimeter, the following formulas can be used:
- Through both sides: p \u003d 2 (a + b).
- Through the area and one of the sides: p \u003d (2s + 2a ²) / a, p \u003d (2s + 2b ²) / b.
The area is a space limited by perimeter.. Three main ways to calculate Square:
- Through the lengths of both sides: s \u003d a * b.
- With perimeter and any one known side: S \u003d (PA - 2 A ²) / 2; S \u003d (Pb - 2 B ²) / 2.
- Diagonally and angle β: s \u003d 0.5 D ² sinβ.
In the tasks of the School Course of Mathematics, it is often necessary to own well properties of diagonals of rectangle. We list the main ones:
- The diagonals are equal to each other and are divided into two equal segments at the point of their intersection.
- The diagonal is defined as the root of the sum of both parties erected into a square (follows from the Pythagores theorem).
- The diagonal separates the rectangle on two triangles with a straight angle.
- The intersection point coincides with the center of the described circle, and the diagonally themselves themselves - with its diameter.
The following formulas are used to calculate the length of the diagonal:
- Using the length and width of the figure: d \u003d √ (a ² + b ²).
- Using the radius of the circle described around the quadricle: D \u003d 2 R.
Definition and properties of the square
Square is a special case of rhombus, a parallelogram or a rectangle. His difference from these figures is that all its corners are straight, and all four sides are equal. Square is the right quadricle.
The quadricle is called the square in the following cases:
- If it is a rectangle, in which length A and width B are equal.
- If it is a rhombus with equal lengths of diagonals and with four straight corners.
The properties of the square include all previously considered properties relating to the rectangle, as well as the following:
- The diagonals are perpendicular to each other (the rhombus property).
- The intersection point coincides with the center of the inscribed circle.
- Both diagonals are divided by a quadrilateral of four identical rectangular and anaid-free triangle.
We give frequently used formulas for 
Calculations of the perimeter, square and square elements:
- Diagonal d \u003d a √2.
- Perimeter p \u003d 4 a.
- Area S \u003d a ².
- The radius of the described circle is twice as smaller than the diagonal: R \u003d 0.5 A √2.
- The radius of the inscribed circle is defined as the half the length of the side: R \u003d A / 2.
Examples of questions and tasks
We will analyze some questions that can be encountered when learning the course of mathematics at school, and solve several simple tasks.
Task 1.. How will the area of \u200b\u200bthe rectangle change, if you increase the length of it three times?
Decision : Denote the area of \u200b\u200bthe original figure S0, and the area of \u200b\u200bthe quadriller with the tripled length of the parties is S1. By the formula discussed earlier, we obtain: S0 \u003d AB. Now increase the length and width 3 times and write: S1 \u003d 3 A 3 B \u003d 9 AB. Comparing S0 and S1, it becomes obvious that the second area is more than the first 9 times.
Question 1. A quadricon with straight corners is a square?
Decision : From the definition it follows that the figure with straight corners is a square only when the lengths of all of its sides are equal. In other cases, the figure is a rectangle.
Task 2.. The diagonal of the rectangle form an angle of 60 degrees. The width of the rectangle is 8. Calculate what is equal to the diagonal.
Decision: Recall that the diagonal of the intersection point is divided by half. Thus, we are dealing with an equilibried triangle with an angle at a vertex of 60 °. Since the triangle is a preceded, then at the base of the angles will also be the same. By easy computing, we obtain that each of them is 60 °. From here it follows that the triangle is equilateral. The width known to us is the basis of the triangle, therefore, half the diagonal is also equal to 8, and the length of the whole diagonal is twice and equal to 16.
Question 2. At the rectangle all parties are equal or not?
Decision : It suffices to remember that all parties should be equal to a square, which is a special occasion of a rectangle. In all other cases, a sufficient condition is the presence of a minimum of 3 straight corners. Equality of the parties is not a mandatory feature.
Task 3.. The square of the square is known and equal to 289. Find the radii inscribed and the circumference described.
Decision :
According to the formulas for the square, we will conduct the following calculations:
- We define what the basic elements of the square are equal: a \u003d √ s \u003d √289 \u003d 17; d \u003d a √2 \u003d 1 7√2.
- We calculate what is equal to the radius of the circumference described around the quadricle: R \u003d 0.5 d \u003d 8.5√2.
- Find the radius of the inscribed circle: R \u003d A / 2 \u003d 17/2 \u003d 8.5.
The rectangle is unique with its simplicity. Based on this figure, students begin to know the foundations of geometry. Therefore, in high schools, not knowing the main properties and signs of the rectangle, in vain, considering this figure is unnecessarily simple.
Rectangle
The definition of the rectangle is known from the elementary school: it is a parallelogram, in which all the corners are equal to 90 degrees. The question arises: what is parallelograms?
Despite the slapper name, this figure is as simple as the rectangle. Parallelogram is a convex quadrilateral, the sides of which are pairwise equal and parallel.
In definition, it is necessary to allocate the word convex. Since convex and non-poor quadrangles are clearly divided into geometry. Moreover, non-poor figures are not studied at all in the school course of mathematics, as they are much more unpredictable in their properties.
Fig. 1. Convex quadrangles
Rectangle is a particular case of a parallelogram. At the same time, there are as other particular cases of a parallelogram, for example, a rhombus; So and other private events of the rectangle are square. Therefore, before proving that the figure is a rectangle, you need to prove that it is a parallelogram.
Properties of rectangles
The properties of the rectangle can be divided into two groups: the properties of the parallelogram and the properties of the rectangle.
Properties Pollogram:
- The opposite sides are pairwise equal and parallel.
- Opposite angles are equal.
Fig. 2. Properties of the parallelogram
Rectangle properties:
- All angles are equal to 90 degrees, which stems from the definition of the figure.
- The diagonal of the rectangle splits the figure into two small equal rectangular triangles. This property is easy to prove. Triangles will be rectangular, as it turns on at one angle of 90 degrees. At the same time, the diagonal will be a common party, and the katenets will be equal, since the opposite sides of the rectangle are pairwise equal and parallel.
- The diagonal of the rectangle is equal.
Fig. 3. Light
Signs of rectangle
The rectangle has only three main features:
- In the corner. If one of the corners of the parallelogram is equal to 90 degrees, then the parallelogram is a rectangle.
- If the three angle of the quadrangle is equal to 90 degrees, then such a quadrangle is a rectangle. Please note that in this case there is no need to prove that under us parallelograms. It is enough to know the values \u200b\u200bof the corners of the quadrangle.
- Diagonal: If the diagonal parallelogram is equal, then such a parallelogram is a rectangle.
Pay attention to which figure is applied by a sign, it matters in evidence.
What is the difference in the sign and properties? The sign is the difference in which you can highlight the figure among others. As a person's name. You see a friend, remember his name and immediately know what to expect from him. But expectations from man are already properties. Properties can be applied only after you have proven that you are one or another figure. And for this proof, it is necessary for signs.
What did we know?
We learned that such a parallelogram. Talked about particular cases of the parallelogram, including the most common - rectangle. Allocated properties and signs of a rectangle. Pay attention to the fact that part of the signs are valid for any quadrilateral, and part only for a parallelogram.
Test on the topic
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