Rectangle elements. What is a rectangle? Special cases of a rectangle
Read also
Rectangle … Spelling dictionary-reference book
Parallelogram, quadrangle, square Dictionary of Russian synonyms. rectangle noun, number of synonyms: 4 square (9) ... Synonym dictionary
Term used in technical analysis market conditions financial markets to indicate price movements that fit into a rectangle on the chart. Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B.. Modern economic dictionary. 2nd ed., revised... Economic dictionary
Dictionary 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 (quadrangle), internal corners which are straight lines, and the 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 Ushakova. D.N. Ushakov. 1935 1940 … Ushakov's Explanatory Dictionary
RECTANGLE, ah, 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). Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary
A type of price movement chart in the form of a triangle, used in technical analysis of financial market conditions. Dictionary of business terms. Akademik.ru. 2001 ... Dictionary of business terms
Books
- Rectangle (+ stickers), Valeria Vilyunova. This sticker book is designed for little readers. At 2 years old, a child enjoys completing exciting tasks by gluing stickers in the right place. This activity is not only...
- Rectangle, Vilyunova V.A. The book “Rectangle” is intended for the youngest readers. With its help, your baby will get acquainted with geometric shapes - rectangle and trapezoid, learn to distinguish and name...
Definition.
Rectangle is a quadrilateral in which two opposite sides are equal and all four angles are equal.The rectangles differ from each other only in the ratio of the long side to the short side, but all four corners are right, that is, 90 degrees.
The long side of a rectangle is called rectangle length, and the short one - 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. The opposite sides of the rectangle have the same length, that is, they are equal:
AB = CD, BC = AD
2. Opposite sides of the rectangle are parallel:
3. The 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:
2d 2 = 2a 2 + 2b 2
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 intersection point:
| AO=BO=CO=DO= | d | ||
| 2 |
10. The point of intersection of the diagonals is called the center of the rectangle and is also the center of the circumcircle
11. The diagonal of a rectangle is the diameter of the circumcircle
12. You can always describe a circle around a rectangle, since the sum of opposite angles is equal to 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 the 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.
Rectangle length is the length of the longer pair of its sides. Rectangle width is the length of the shorter pair of its sides.Formulas for determining the lengths of the sides of a rectangle
1. Formula for the side of a rectangle (length and width of the rectangle) through the diagonal and the other side:
a = √ d 2 - b 2
b = √ d 2 - a 2
2. Formula for the side of a rectangle (length and width of the rectangle) through the area and the other side:
| b = dcos | β |
| 2 |
Diagonal of a rectangle
Definition.
Diagonal rectangle Any segment connecting two vertices of opposite corners of a rectangle is called.Formulas for determining the length of the diagonal of a rectangle
1. Formula for the diagonal of a rectangle using two sides of the rectangle (via the Pythagorean theorem):
d = √ a 2 + b 2
2. Formula for the diagonal of a rectangle using the area and any side:
4. Formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:
d = 2R
5. Formula for the diagonal of a rectangle in terms of the diameter of the circumcircle:
d = D o
6. Formula for the diagonal of a rectangle using the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:
8. Formula for the diagonal of a rectangle through sine acute angle between the diagonals and the area of the rectangle
d = √2S: sin β
Perimeter of a rectangle
Definition.
Perimeter of a rectangle is the sum of the lengths of all sides of a rectangle.Formulas for determining the length of the perimeter of a rectangle
1. Formula for the perimeter of a rectangle using two sides of the rectangle:
P = 2a + 2b
P = 2(a + b)
2. Formula for the perimeter of a rectangle using area and any side:
| P= | 2S + 2a 2 | = | 2S + 2b 2 |
| a | b |
3. Formula for the perimeter of a rectangle using the diagonal and any side:
P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)
4. Formula for the perimeter of a rectangle using the radius of the circumcircle and any side:
P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)
5. Formula for the perimeter of a rectangle using the diameter of the circumscribed circle and any side:
P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)
Area of a rectangle
Definition.
Area of a rectangle called the space limited by the sides of the rectangle, that is, within the perimeter of the rectangle.Formulas for determining the area of a rectangle
1. Formula for the area of a rectangle using two sides:
S = a b
2. Formula for the area of a rectangle using the perimeter and any side:
5. Formula for the area of a rectangle using the radius of the circumscribed circle and any side:
S = a √4R 2 - a 2= b √4R 2 - b 2
6. Formula for the area of a rectangle using the diameter of the circumcircle and any side:
S = a √D o 2 - a 2= b √D o 2 - b 2
Circle circumscribed around a rectangle
Definition.
A circle circumscribed around a rectangle is a circle passing through the four vertices of a rectangle, the center of which lies at the intersection of the diagonals of the rectangle.Formulas for determining the radius of a circle circumscribed around a rectangle
1. Formula for the radius of a circle circumscribed around a rectangle through two sides:
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.
A 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 y geometric figure 3 of the 4 angles are 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
have equal length; 3 – all angles are equal to 90 degrees. If at least one condition is met, the figure can be called a rectangle.
The video course “Get an A” includes all the topics necessary for successful
passing the Unified State Exam in mathematics for 60-65 points. Completely all tasks 1-13 of the Profile Unified State Exam in mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes! Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.
All the necessary theory. big topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.
Hundreds of Unified State Exam tasks. Word problems and probability theory. Simple and easy to remember algorithms for solving problems. Geometry. Theory, reference material, analysis of all types of Unified State Examination tasks. Stereometry. Tricky Tricks solutions, useful cheat sheets, development of spatial imagination. Trigonometry from scratch to problem 13. Understanding instead of cramming. Clear explanations of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Basis for solution complex tasks 2 parts of the Unified State Exam.