Five essential properties of the concept of a rectangle. What is a rectangle? Special cases of a rectangle

A rectangle is a parallelogram in which all angles are right angles (equal to 90 degrees). The area of ​​a rectangle is equal to the product of its adjacent sides. The diagonals of a rectangle are equal. The second formula for finding the area of ​​a rectangle comes from the formula for the area of ​​a quadrilateral in terms of diagonals.

Rectangle is a quadrilateral in which every corner is a right angle.

A square is a special case of a rectangle.

A rectangle has two pairs of equal sides. The length of the longest pair of sides is called rectangle length, and the length of the shortest - rectangle width.

Rectangle properties

1. A rectangle is a parallelogram.

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.

\(AC = BD\)

According to property 1 the rectangle is a parallelogram, which means \(AB = CD \) .

Consequently, \(\triangle ABD = \triangle DCA \) 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 \(AC = BD \) .

Only a rectangle of all figures (only from parallelograms!) Has equal diagonals.

Let's prove this too.

\(\Rightarrow AB = CD \) , \(AC = BD \) 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 \(\angle A = \angle B = \angle C = \angle D \). All of them by \(90^(\circ) \) . The sum is \(360^(\circ) \) .

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.

\(AO = BO = CO = DO \)

9. The point of intersection of the diagonals is the center of the rectangle and the circumscribed circle.

Definition.

Rectangles differ from each other only in the ratio of the long side to the short side, but all four of them are right, that is, 90 degrees each.

The long side of a rectangle is called rectangle length, and the short rectangle width.

The sides of a rectangle are also its heights.

Basic properties of a rectangle

A rectangle can be a parallelogram, a square or a rhombus.

1. Opposite sides of a rectangle have the same length, that is, they are equal:

AB=CD, BC=AD

2. Opposite sides of the rectangle are parallel:

3. Adjacent sides of a rectangle are always perpendicular:

AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB

4. All four corners of the rectangle are straight:

∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°

5. The sum of the angles of a rectangle is 360 degrees:

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

6. The diagonals of a rectangle have the same length:

7. The sum of the squares of the diagonal of a rectangle is equal to the sum of the squares of the sides:

2d2 = 2a2 + 2b2

8. Each diagonal of a rectangle divides the rectangle into two identical figures, namely right triangles.

9. The diagonals of the rectangle intersect and are divided in half at the point of intersection:

10. The intersection point of the diagonals is called the center of the rectangle and is also the center of the circumscribed circle

11. The diagonal of a rectangle is the diameter of the circumscribed circle

12. A circle can always be described around a rectangle, since the sum of opposite angles is 180 degrees:

∠ABC = ∠CDA = 180° ∠BCD = ∠DAB = 180°

13. A circle cannot be inscribed in a rectangle whose length is not equal to its width, since the sums of opposite sides are not equal to each other (a circle can only be inscribed in a special case of a rectangle - a square).

Sides of a rectangle

Definition.

Formulas for determining the lengths of the sides of a rectangle

1. The formula for the side of a rectangle (the length and width of the rectangle) in terms of the diagonal and the other side:

a = √ d 2 - b 2

b = √ d 2 - a 2

2. The formula for the side of a rectangle (the length and width of the rectangle) in terms of the area and the other side:

Rectangle Diagonal

Definition.

Formulas for determining the length of the diagonal of a rectangle

1. The formula for the diagonal of a rectangle in terms of two sides of the rectangle (via the Pythagorean theorem):

d = √ a 2 + b 2

2. The formula for the diagonal of a rectangle in terms of area and any side:

4. The formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:

d=2R

5. The formula for the diagonal of a rectangle in terms of the diameter of the circumscribed circle:

d = D o

6. The formula of the diagonal of a rectangle in terms of the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:

8. The formula of the diagonal of a rectangle in terms of the sine of an acute angle between the diagonals and the area of ​​the rectangle

d = √2S: sinβ

Perimeter of a rectangle

Definition.

Formulas for determining the length of the perimeter of a rectangle

1. The formula for the perimeter of a rectangle in terms of two sides of the rectangle:

P = 2a + 2b

P = 2(a+b)

2. The formula for the perimeter of a rectangle in terms of area and any side:

3. Formula for the perimeter of a rectangle in terms of the diagonal and any side:

P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)

4. The formula for the perimeter of a rectangle in terms of the radius of the circumscribed circle and any side:

P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)

5. The formula for the perimeter of a rectangle in terms of the diameter of the circumscribed circle and any side:

P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)

Rectangle area

Definition.

Formulas for determining the area of ​​a rectangle

1. The formula for the area of ​​a rectangle in terms of two sides:

S = a b

2. The formula for the area of ​​a rectangle through the perimeter and any side:

5. The formula for the area of ​​a rectangle in terms of the radius of the circumscribed circle and any side:

S = a √4R 2 - a 2= b √4R 2 - b 2

6. The formula for the area of ​​a rectangle in terms of the diameter of the circumscribed circle and any side:

S \u003d a √ D o 2 - a 2= b √ D o 2 - b 2

Circle circumscribed around a rectangle

Definition.

Formulas for determining the radius of a circle circumscribed around a rectangle

1. The formula for the radius of a circle circumscribed around a rectangle through two sides:

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;
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;
Cultivate interest in 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 quadrangle 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 geometric figures, which refer 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;
A rectangle has equal opposite sides and angles;
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 triangle;
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.

But 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. French writer humorist Alphonse Allais produced a whole series of his works and eventually 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 larger side which is equal to the golden ratio to the smaller one.

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 from 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 the usual, standard form of the house, symbolizing the woman who owns this house.

Rectangle 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

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).

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.

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.