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An equation with a modulus on a square-rhombus plane. Rhombus as a geometric figure. The diagonals of a rhombus are the bisectors of its angles

And again the question is: is a rhombus a parallelogram or not?

With full right - a parallelogram, because it has and (remember our sign 2).

And again, since a rhombus is a parallelogram, then it must have all the properties of a parallelogram. This means that a rhombus has opposite angles equal, opposite sides are parallel, and the diagonals are bisected by the point of intersection.

Rhombus Properties

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 we have not just a parallelogram, but a rhombus.

Signs of a rhombus

And pay attention again: there should be not just a quadrangle with perpendicular diagonals, but a parallelogram. Make sure:

No, of course not, although its diagonals and are perpendicular, and the diagonal is the bisector of angles u. But ... the diagonals do not divide, the intersection point in half, 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 comes out of this.

Is it clear why? - rhombus - the bisector of angle A, which is equal to. So it divides (and also) into two angles along.

Well, it's quite clear: the rectangle's diagonals are equal; rhombus diagonals are perpendicular, and in general - parallelogram diagonals are divided by the point of intersection in half.

AVERAGE LEVEL

Properties of quadrilaterals. Parallelogram

Parallelogram Properties

Attention! Words " parallelogram properties» means that if you have a task eat parallelogram, then all of the following can be used.

Theorem on the properties of a parallelogram.

In any parallelogram:

Let's see why this is true, in other words WE WILL PROVE theorem.

So why is 1) true?

Since it is a parallelogram, then:

  • like lying crosswise
  • as lying across.

Hence, (on the II basis: and - general.)

Well, once, then - that's it! - proved.

But by the way! We also proved 2)!

Why? But after all (look at the picture), that is, namely, because.

Only 3 left).

To do this, you still have to draw a second diagonal.

And now we see that - according to the II sign (the angle and the side "between" them).

Properties proven! Let's move on to the signs.

Parallelogram features

Recall that the sign of a parallelogram answers the question "how to find out?" That the 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, that's even easier! Let's draw a diagonal again.

Which means:

AND is also easy. But… different!

Means, . Wow! But also - internal one-sided at a secant!

Therefore the fact that means that.

And if you look from the other side, then they are internal one-sided at a secant! And therefore.

See how great it is?!

And again simply:

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

So, on two legs (and - general).

Well, since the triangles are equal, then their hypotenuses are also equal.

Proved that!

And imagine, the equality of the diagonals is a distinctive property of a rectangle among all parallelograms. That is, the following statement is true

Let's see why?

So, (meaning the angles of the parallelogram). But once again, remember that - a parallelogram, and therefore.

Means, . And, of course, it follows from this that each of them After all, in the amount they should give!

Here we have proved that if parallelogram suddenly (!) will be equal diagonals, then this exactly a rectangle.

But! Pay attention! This is about parallelograms! Not any a quadrilateral with equal diagonals is a rectangle, and only parallelogram!

Properties of quadrilaterals. Rhombus

And again the question is: is a rhombus a parallelogram or not?

With full right - a parallelogram, because it has and (Remember our sign 2).

And again, since a rhombus is a parallelogram, it must have all the properties of a parallelogram. This means that a rhombus has opposite angles equal, opposite sides are parallel, and the diagonals are bisected by the point of intersection.

But there are also special properties. We formulate.

Rhombus Properties

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 and turned out to be the 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.

Rhombus signs.

Why is that? And look

Hence, and both these triangles are isosceles.

In order to be a rhombus, a quadrilateral must first "become" a parallelogram, and then already demonstrate 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 comes out of this.

Is it clear why? Square - rhombus - the bisector of the angle, which is equal to. So it divides (and also) into two angles along.

Well, it's quite clear: the rectangle's diagonals are equal; rhombus diagonals are perpendicular, and in general - parallelogram diagonals are divided by the point of intersection in half.

Why? Well, just apply the Pythagorean Theorem to.

SUMMARY AND BASIC FORMULA

Parallelogram properties:

  1. Opposite sides are equal: , .
  2. Opposite angles are: , .
  3. The angles at one side add up to: , .
  4. The diagonals are divided by the intersection point in half: .

Rectangle properties:

  1. The diagonals of a rectangle are: .
  2. Rectangle is a parallelogram (all properties of a parallelogram are fulfilled for a rectangle).

Rhombus properties:

  1. The diagonals of the rhombus are perpendicular: .
  2. The diagonals of a rhombus are the bisectors of its angles: ; ; ; .
  3. A rhombus is a parallelogram (all properties of a parallelogram are fulfilled for a rhombus).

Square properties:

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. As well as.

summary of other presentations

"Tasks for signs of similarity of triangles" - Similarity of triangles. Determining the height of an object using a mirror. Determining the height of an object from a puddle. Solution of practical problems. The shadow of the stick. Determining the height of an object. Measuring the height of large objects. Lesson motto. Solution of problems according to ready-made drawings. Independent work. Gymnastics for the eyes. Thales method. Individual card. Determining the height of the pyramid. Name similar triangles.

"Properties of quadrilaterals" - Names of quadrilaterals. All angles are right. Properties of quadrilaterals. Trapeze. A square is a rectangle in which all sides are equal. Elements of a parallelogram. Diagonals bisect the corners. Quadrilateral. Dictation. Diagonal. opposite corners. Help Dunno fix the deuce. Historical information. Quadrilaterals and their properties. Diagonals. Rhombus. opposite sides. Parties.

"Rhombus" - Signs. Perimeter. The appearance of the rhombus. Rhombus story. Rhombus. A rhombus with diagonals. What is a rhombus. Area formula. Interesting Facts. Rhombus properties. Rhombus in life.

"The solution of the Pythagorean theorem" - Proof by decomposition. Square area. The simplest proof. Proof of Perigal. Pythagoreans. Diagonal. 9th century CE proof followers. Height. Diameter. Complete proof. Motive. Hexagons. Proof by subtraction. Square. Rectangle. Possibilities of application of the theorem. Gutheil's proof. Application of the theorem. Lotus problem. History of the theorem.

"Area of ​​a rectangle" grade 8 "- The area of ​​\u200b\u200bthe square is equal to the square of its side. Area. Find the area and perimeter of the square. Area units. A polygon is made up of several polygons. Find the area of ​​a triangle. The sides of each of the rectangles. Units. Find the area of ​​the square. ABCD and DSMK are squares. The area of ​​a rhombus is half the product of its diagonals. A parallelogram is drawn on side AB. Find the area of ​​the hexagon.

"Trapezoid" Grade 8 "- The trapezius muscles of both sides of the back together have the shape of a trapezoid. Tasks for oral work. Are quadrilaterals trapezoids. Properties of an isosceles trapezoid. Signs of an isosceles trapezoid. Types of trapezoid. The area of ​​the trapezoid. Trapeze elements. Definition. The median line of the trapezium. Trapeze. The geometric figure was named so for its resemblance to a small table.

with equal sides. A rhombus with right angles is square .

A rhombus is considered as a kind of parallelogram, with two adjacent equal sides, either with mutually perpendicular diagonals, or with diagonals dividing the angle into 2 equal parts.

Rhombus properties.

1. Rhombus is a parallelogram, so opposite sides are the same length and parallel in pairs, AB || CD, AD || Sun.

2. Angle of intersection of diagonals rhombus is straight (AC⊥ BD) and the intersection point are divided into two identical parts. That is, the diagonals divide the rhombus into 4 triangles - rectangular.

3. Rhombus diagonals are the bisectors of its angles (∠ DCA=∠ bca,∠ ABD=∠ CBD etc. ).

4. Sum of the squares of the diagonals equals the square of the side multiplied by four (derived from the parallelogram identity).

Rhombus signs.

Parallelogram ABCD will be called a rhombus only if at least one of the following conditions is met:

1. 2 of its adjacent sides are the same length (that is, all sides of a rhombus are equal, AB=BC=CD=AD).

2. The angle of intersection of the diagonals of the straight line ( AC⊥ BD).

3. A 1-on of diagonals bisects the corners that contain it.

Suppose we do not know in advance that the quadrilateral turns out to be a parallelogram, but it is known that all its sides are equal. So this quadrilateral is a rhombus.

Rhombus symmetry.

Rhombus is symmetrical relative to all its diagonals, it is often used in ornaments and parquets.

The perimeter of a rhombus.

The perimeter of a geometric figure- the total length of the boundaries of a flat geometric figure. The perimeter has the same dimension as the length.

Rhombus is one of the simplest geometric shapes. We so often meet with a rhombus in geometric problems that the words "fantastic" and "rhombus" seem to us incompatible concepts. Meanwhile, the amazing, as they say, is nearby ... in Britain. But first, let's remember what a "rhombus" is, its signs and properties.

The term "rhombus" in translation from ancient Greek means "tambourine". And this is no coincidence. And here's the thing. A tambourine at least once in a lifetime, but everyone saw it. And everyone knows that it is round. But a long time ago, tambourines were made just in the shape of a square or a rhombus. Moreover, the name of the diamond suit is also associated with this fact.

From geometry, we imagine what a rhombus looks like. This is a quadrilateral, which is depicted as an inclined square. But in no case should you confuse a rhombus and a square. It is more correct to say that a rhombus is a special case of a parallelogram. The only difference is that all sides of the rhombus are equal. To quickly and correctly solve problems in geometry, you need to remember the properties of a rhombus. By the way, a rhombus has all the properties of a parallelogram. So:

Rhombus properties:

  1. opposite sides are equal;
  2. opposite angles are equal;
  3. the diagonals of the rhombus intersect under the straight line and are divided in half at the point of intersection;
  4. the sum of the angles adjacent to one side is 180°;
  5. the sum of the squares of the diagonals is equal to the sum of the squares of all the sides;
  6. the diagonals are the bisectors of its angles.

Signs of a rhombus:

  1. if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus;
  2. if the diagonal of a parallelogram is the bisector of its angle, then the parallelogram is a rhombus.

And one more important point, without the knowledge of which it is not possible to successfully solve the problem - formulas. Below are the formulas for finding the area of ​​any rhombus, which are used depending on the known data: height, diagonal, side, radius of the inscribed circle. In the following formulas, symbols are used: a - side of the rhombus, h a - height drawn to side a, but- the angle between the sides, d 1 d 2 - the diagonals of the rhombus.

Basic formulas:

S = a 2 sin but

S = 1/2 (d 1 d 2)

S = 4r2 / sin a

There is another formula that is not used so often, but is useful:

d 1 2 + d 2 2 = 4a 2 or the sum of the squares of the diagonals is equal to the square of the side times 4.

And now it's time to go back to the very beginning. What is so amazing maybe in this figurine? It turns out that in the 19th century, during archaeological excavations, a rhombus was found. Yes, not simple, but golden, and, in the truest sense of the word! This find from the British mound Bash was found in the Wilsford area, not far from the famous Stonehenge. The mysterious rhombus is a polished plate on which unusual patterns are engraved. Its size is 15.2 x 17.8 cm (rhombus with only a small reservation). In addition to the edging, the plate has three more smaller diamond-shaped patterns, which are supposedly nested into each other. At the same time, a rhombic grid is engraved in the center of the latter. Along the edges of the rhombus is a chevron pattern - nine characters on each side of the rhombus. There are thirty-six such triangles in total.

Of course, this product is very expensive, but it is also obvious that the creation of such a rhombus pursued a specific goal. That's just what, scientists could not figure out for a long time.

One of the more plausible and accepted versions concerns Stonehenge directly. It is known that the constructions of Stonehenge were erected gradually, over several centuries. It is believed that construction began around 3000 BC. It should be noted that gold in Britain became known already somewhere from 2800 BC. From this it can be assumed that the golden rhombus could well have been a priest's tool. In particular, the vizier. Such a hypothesis was brought to the attention of modern scientists by Professor A. Tom, a well-known researcher of Stonehenge, in the last quarter of the 20th century.

Not everyone can imagine that the ancient builders could accurately determine the angles on the ground. Nevertheless, the English researcher D. Furlong proposed a method that, in his opinion, the ancient Egyptians could use. Furlong believed that our ancestors used pre-selected aspect ratios in right triangles. After all, it has long been known that the Egyptians widely used a triangle with sides of three, four and five dimensional units. Apparently, the ancient inhabitants of the British Isles also knew many such tricks.

Well, even if you imagine that the people who built Stonehenge were excellent surveyors, how could a golden diamond help them in this? Hardly any modern surveyor will be able to answer this question. Most likely, the fact that Furlong was a surveyor by profession, gave him the opportunity to solve this riddle. After careful study, the researcher came to the conclusion that the polished gold rhombus with markings is excellent for use as a reflector of sunlight, in other words, a special dimensional mirror.

It was proved that in order to quickly determine the azimuth on the ground with fairly small errors, it was necessary to use two similar mirrors. The scheme was as follows: one priest, for example, stood on the top of one hill, and the other in the adjacent valley. It was also necessary to pre-set the distance between the priests. This can be done with just a few steps. Although they usually used a measuring stick, since the results were more reliable. Two diamond-shaped metal mirrors provide a right angle. And then it is easy to measure almost any required angles. D. Furlong even gave a table of such pairs of integers, which allows you to set any angle with an error of one degree. It is most likely that the priests of the Stonehenge era used this method. Of course, to confirm this hypothesis, it would be necessary to find a second, paired golden rhombus, but, apparently, this is not worth it. After all, the evidence is quite clear. In addition to calculating azimuths on the ground, another ability of an amazing golden rhombus was discovered. This amazing little thing is allowed to calculate the moments of the winter and summer solstices, spring and autumn equinoxes. This was an indispensable quality for the life of the ancient Egyptians, who worshiped then primarily the Sun.

It is likely that the imposing appearance of the rhombus was not only an indispensable tool for the priests, but was also a spectacular decoration for its owner. Generally speaking, the vast majority of jewelry found at first glance, expensive today, are, as it turns out later, measuring instruments.

So people have always been drawn to the unknown. And, judging by the fact that so much remains mysterious and unproven in our world, a person will try to find the clues to antiquity for a long time to come. And it's very cool! After all, we can learn a lot from our ancestors. To do this, you need to know a lot, be able to learn and learn. But it is impossible to become such a highly qualified specialist without basic knowledge. After all, after all, every great archaeologist, discoverer once went to school!

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AB \parallel CD,\;BC \parallel AD

AB=CD,\;BC=AD

2. The diagonals of the rhombus are perpendicular.

AC\perp BD

Proof

Since a rhombus is a parallelogram, its diagonals are bisected.

So \triangle BOC = \triangle DOC on three sides (BO = OD , OC is joint, BC = CD ). We get that \angle BOC = \angle COD , and they are adjacent.

\Rightarrow \angle BOC = 90^(\circ) and \angle COD = 90^(\circ) .

3. The intersection point of the diagonals bisects them.

AC=2\cdot AO=2\cdot CO

BD=2\cdot BO=2\cdot DO

4. The diagonals of a rhombus are the bisectors of its angles.

\angle1 = \angle2; \; \angle 5 = \angle 6;

\angle 3 = \angle 4; \; \angle 7 = \angle 8.

Proof

Due to the fact that the diagonals are divided by the intersection point in half, and all sides of the rhombus are equal to each other, the whole figure is divided by the diagonals into 4 equal triangles:

\triangle BOC, \; \triangle BOA, \; \triangle AOD, \; \triangle COD.

This means that BD , AC are bisectors.

5. Diagonals form 4 right-angled triangles from a rhombus.

6. Any rhombus can contain a circle centered at the point of intersection of its diagonals.

7. The sum of the squares of the diagonals is equal to the square of one of the sides of the rhombus multiplied by four

AC^2 + BD^2 = 4\cdot AB^2

Signs of a rhombus

1. A parallelogram with perpendicular diagonals is a rhombus.

\begin(cases) AC \perp BD \\ ABCD \end(cases)- parallelogram, \Rightarrow ABCD - rhombus.

Proof

ABCD is a parallelogram \Rightarrow AO = CO ; BO=OD. It is also indicated that AC \perp BD \Rightarrow \triangle AOB = \triangle BOC = \triangle COD = \triangle AOD- on 2 legs.

It turns out that AB = BC = CD = AD.

Proven!

2. When in a parallelogram at least one of the diagonals divides both angles (through which it passes) in half, then this figure will be a rhombus.

Proof

On a note: not every figure (quadrilateral) with perpendicular diagonals will be a rhombus.

For instance:

This is no longer a rhombus, despite the perpendicularity of the diagonals.

To distinguish it, it is worth remembering that at first the quadrilateral must be a parallelogram and have